Contents

Elliptic curves over function fields

Douglas Ulmer 213

Elliptic curves over function fields 215

Introduction 215

Lecture 0. Background on curves and function fields 217

1. Terminology 217
2. Function fields and curves 217
3. Zeta functions 218
4. Cohomology 219
5. Jacobians 219
6. Tate’s theorem on homomorphisms of abelian varieties 221

Lecture 1. Elliptic curves over function fields 223

1. Elliptic curves 223
2. Frobenius 224
3. The Hasse invariant 225
4. Endomorphisms 226
5. The Mordell-Weil-Lang-Néron theorem 226
6. The constant case 227
7. Torsion 228
8. Local invariants 230
9. The L-function 231
10. The basic BSD conjecture 232
11. The Tate-Shafarevich group 232
12. Statements of the main results 233
13. The rest of the course 234

Lecture 2. Surfaces and the Tate conjecture 237

1. Motivation 237
2. Surfaces 237
3. Divisors and the Néron-Severi group 238
4. The Picard scheme 239
5. Intersection numbers and numerical equivalence 239
6. Cycle classes and homological equivalence 240
7. Comparison of equivalence relations on divisors 241
8. Examples 241
9. Tate’s conjectures T1 and T2 243
10. T1 and the Brauer group 244
i
ii LECTURE 0. CONTENTS

11. The descent property of T1 246

12. Tate’s theorem on products 246
13. Products of curves and DPC 247
Lecture 3. Elliptic curves and elliptic surfaces 249
1. Curves and surfaces 249
2. The bundle ω and the height of E 252
3. Examples 252
4. E and the classification of surfaces 254
5. Points and divisors, Shioda-Tate 255
6. L-functions and Zeta-functions 256
7. The Tate-Shafarevich and Brauer groups 257
8. The main classical results 258
9. Domination by a product of curves 259
10. Four monomials 259
11. Berger’s construction 260
Lecture 4. Unbounded ranks in towers 263
1. Grothendieck’s analysis of L-functions 263
2. The case of an elliptic curve 266
3. Large analytic ranks in towers 267
4. Large algebraic ranks 270
Lecture 5. More applications of products of curves 273
1. More on Berger’s construction 273
2. A rank formula 274
3. First examples 275
4. Explicit points 276
5. Another example 277
6. Further developments 277

Bibliography 279
Elliptic curves over function fields

Douglas Ulmer
IAS/Park City Mathematics Series
Volume XX, XXXX

Elliptic curves over function fields

Douglas Ulmer

Introduction
These are the notes from a course of five lectures at the 2009 Park City Math
Institute. The focus is on elliptic curves over function fields over finite fields. In
the first three lectures, we explain the main classical results (mainly due to Tate)
on the Birch and Swinnerton-Dyer conjecture in this context and its connection to
the Tate conjecture about divisors on surfaces. This is preceded by a “Lecture 0”
on background material. In the remaining two lectures, we discuss more recent
developments on elliptic curves of large rank and constructions of explicit points in
high rank situations.
A great deal of this material generalizes naturally to the context of curves and
Jacobians of any genus over function fields over arbitrary ground fields. These gen-
eralizations were discussed in a course of 12 lectures at the CRM in Barcelona
in February, 2010, and will be written up as a companion to these notes, see
[Ulm11]. Unfortunately, theorems on unbounded ranks over function fields are
currently known only in the context of finite ground fields.
Finally, we mention here that very interesting theorems of Gross-Zagier type
exist also in the function field context. These would be the subject of another series
of lectures and we will not say anything more about them in these notes.
It is a pleasure to thank the organizers of the 2009 PCMI for the invitation to
speak, the students for their interest, enthusiasm, and stimulating questions, and
the “elder statesmen”—Bryan Birch, Dick Gross, John Tate, and Yuri Zarhin—
for their remarks and encouragement. Thanks also to Keith Conrad for bringing
the fascinating historical articles of Roquette [Roq06] to my attention. Last but
not least, thanks are due as well to Lisa Berger, Tommy Occhipinti, Karl Rubin,
Alice Silverberg, Yuri Zarhin, and an anonymous referee for their suggestions and
TEXnical advice.

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332

E-mail address: douglas.ulmer@math.gatech.edu

c 2012 American Mathematical Society

215
 LECTURE 0

Background on curves and function fields

This “Lecture 0” covers definitions and notations that are probably familiar
to many readers and that were reviewed very quickly during the PCMI lectures.
Readers are invited to skip it and refer back as necessary.

1. Terminology
Throughout, we use the language of schemes. This is necessary to be on firm ground
when dealing with some of the more subtle aspects involving non-perfect ground
fields and possibly non-reduced group schemes. However, the instances where we
use any hard results from this theory are isolated and students should be able to
follow readily the main lines of discussion, perhaps with the assistance of a friendly
algebraic geometer.
Throughout, a variety over a field F is a separated, reduced scheme of finite
type over Spec F . A curve is a variety purely of dimension 1 and a surface is a
variety purely of dimension 2.

2. Function fields and curves

Throughout, p will be a prime number and Fq will denote the field with q elements
with q a power of p. We write C for a smooth, projective, and absolutely irreducible
curve of genus g over Fq and we write K = Fq (C) for the function field of C over
Fq . The most important example is when C = P1 , the projective line, in which case
K = Fq (C) = Fq (t) is the field of rational functions in a variable t over Fq .
We write v for a closed point of C, or equivalently for an equivalence class of
valuations of K. For each such v we write O(v) for the local ring at v (the ring
of rational functions on C regular at v), mv ⊂ O(v) for the maximal ideal (those
functions vanishing at v), and κv = O(v) /mv for the residue field at v. The extension
κv /Fq is finite and we set deg(v) = [κv : Fq ] and qv = q deg(v) so that κv ∼
= Fqv .
For example, in the case where C = P1 , the “finite” places of C correspond
bijectively to monic irreducible polynomials f ∈ Fq [t]. If v corresponds to f , then
O(v) is the set of ratios g/h where g, h ∈ Fq [t] and f does not divide h. The maximal
ideal mv consists of ratios g/h where f does divide g, and the degree of v is the
degree of f as a polynomial in t. There is one more place of K, the “infinite” place
v = ∞. The local ring consists of ratios g/h with g, h ∈ Fq [t] and deg(g) ≤ deg(h).
The maximal ideal consists of ratios g/h where deg(g) < deg(h) and the degree of
v = ∞ is 1. The finite and infinite places of P1 give all closed points of P1 .
We write K sep for a separable closure of K and let GK = Gal(K sep /K). We
write Fq for the algebraic closure of Fq in K sep . For each place v of K we have the
decomposition group Dv (defined only up to conjugacy), its normal subgroup the
inertia group Iv ⊂ Dv , and Frv the (geometric) Frobenius at v, a canonical generator
217
218 LECTURE 0. BACKGROUND ON CURVES AND FUNCTION FIELDS

−1
of the quotient Dv /Iv ∼
= Gal(Fq /Fq ) that acts as x 7→ xqv on the residue field at
a place w dividing v in a finite extension F ⊂ K sep unramified over v.
General references for this section and the next are [Gol03], [Ros02], and [Sti09].

3. Zeta functions
Let X be a variety over the finite field Fq . Extending the notation of the previous
section, if x is a closed point of X , we write κx for the residue field at x, qx for its
cardinality, and deg(x) for [κx : Fq ].
We define the Z and ζ functions of X via Euler products:
Y −1
Z(X , T ) = 1 − T deg(x)
x
and Y
−1
ζ(X , s) = Z(X , q −s ) = 1 − qx−s
x
where the products are over the closed points of X . It is a standard exercise to
show that  
n
X T
Z(X , T ) = exp  Nn 
n
n≥1

where Nn is the number of Fqn -valued points of X . It follows from a crude estimate
for the number of Fqn points of X that the Euler product defining ζ(X , s) converges
in the half plane Re(s) > dim X .
If X is smooth and projective, then it is known that Z(X , T ) is a rational
function of the form Qdim X −1
i=0 P2i+1 (T )
Qdim X
i=0 P2i (T )
where P0 (T ) = (1 − T ), P2 dim X (T ) = (1 − q dim X T ), and for all 0 ≤ i ≤ 2 dim X
Pi (T ) is a polynomial with integer coefficients and constant term 1. We denote the
inverse roots of Pi by αij so that
Y
Pi (T ) = (1 − αij T )
j

The inverse roots αij of Pi (T ) are algebraic integers that have absolute value
q i/2 in every complex embedding. (We say that they are Weil numbers of size
q i/2 .) It follows that ζ(X , s) has a meromorphic continuation to the whole s
plane, with poles on the lines Re s ∈ {0, . . . , dim X } and zeroes on the lines Re s ∈
{1/2, . . . , dim X −1/2}. This is the analogue of the Riemann hypothesis for ζ(X , s).
It is also known that the set of inverse roots of Pi (T ) (with multiplicities) is
stable under αij 7→ q i /αij and that for i ≤ dim X , P2d−i (T ) = Pi (q d−i T ). Thus
ζ(X , s) satisfies a functional equation when s is replaced by dim X − s.
In the case where X is a curve, P1 (T ) has degree 2g (g = the genus of C) and
has the form
2g
Y
P1 (T ) = 1 + · · · + q g T 2g = (1 − α1j T ).
j=1
2πi 2πi
Thus ζ(C, s) has simple poles for s ∈ log q Z and s ∈ 1 + log q Z and its zeroes lie on
the line Re s = 1/2.
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 219

For a fascinating history of the early work on zeta functions and the Riemann
hypothesis for curves over finite fields, see [Roq06] and parts I and II of that work.

4. Cohomology
Assume that X is a smooth projective variety over k = Fq . We write X for X ×Fq Fq .
Note that Gk = Gal(Fq /Fq ) acts on X via the factor Fq .
Choose a prime ` 6= p. We have `-adic cohomology groups H i (X , Q` ) which
are finite-dimensional Q` -vector spaces and which vanish unless 0 ≤ i ≤ 2 dim X .
Functoriality in X gives a continuous action of Gal(Fq /Fq ). Since the geometric
−1
Frobenius (Frq (a) = aq ) is a topological generator of Gal(Fq /Fq ), the character-
istic polynomial of Frq on H i (X , Q` ) determines the eigenvalues of the action of
Gal(Fq /Fq ); in fancier language, it determines the action up to semi-simplification.
An important result (inspired by [Wei49] and proven in great generality in
[SGA5]) says that the factors Pi of Z(X , t) are characteristic polynomials of Frobe-
nius:
(4.1) Pi (T ) = det(1 − T Frq |H i (X , Q` )).
From this point of view, the functional equation and Riemann hypothesis for
Z(X , T ) are statements about duality and purity.
To discuss the connections, we need more notation. Let Z` (1) = limn µ`n (Fq )
←−
and Q` (1) = Z` (1) ⊗Z` Q` , so that Q` (1) is a one-dimensional Q` -vector space on
which Gal(Fq /Fq ) acts via the `-adic cyclotomic character. More generally, for
n > 0 set Q` (n) = Q` (1)⊗n (n-th tensor power) and Q` (−n) = Hom(Q` (n), Q` ),
so that for all n, Q` (n) is a one-dimensional Q` -vector space on which Gal(Fq /Fq )
acts via the nth power of the `-adic cyclotomic character.
We have H 0 (X , Q` ) ∼ = Q` (with trivial Galois action) and H 2 dim X (X , Q` ) ∼
=
Q` (dim X ). The functional equation follows from the fact that we have a canonical
non-degenerate, Galois equivariant pairing
H i (X , Q` ) × H 2 dim X −i (X , Q` ) → H 2 dim X (X , Q` ) ∼
= Q` (dim X ).
Indeed, the non-degeneracy of this pairing implies that if α is an eigenvalue of Frq
on H i (X , Q` ), then q dim X /α is an eigenvalue of Frq on H 2 dim X −i (X , Q` ).
The Riemann hypothesis in this context is the statement that the eigenvalues
of Frq on H i (X , Q` ) are algebraic integers with absolute value q i/2 in every complex
embedding.
See [SGA4 12 ] or [Mil80] for an overview of étale cohomology and its connections
with the Weil conjectures.

5. Jacobians
5.1. Picard and Albanese properties
We briefly review two (dual) universal properties of the Jacobian of a curve that
we will need. See [Mil86b] for more details.
We assume throughout that the curve C has an Fq -rational point x, i.e., a
closed point with residue field Fq . If T is another connected variety over Fq with
an Fq -rational point t, a divisorial correspondence between (C, x) and (T, t) is an
invertible sheaf L on C ×Fq T such that L|C×t and L|x×T are trivial. Two divisorial
correspondences are equal when they are isomorphic as invertible sheaves. Note
220 LECTURE 0. BACKGROUND ON CURVES AND FUNCTION FIELDS

that the set of divisorial correspondences between (C, x) and (T, t) forms a group
under tensor product and is thus a subgroup of Pic(C × T ). We write
DivCorr((C, x), (T, t)) ⊂ Pic(C × T )
for this subgroup. One may think of a divisorial correspondence as giving a family
of invertible sheaves on C: s 7→ L|C×s .
Let J = JC be the Jacobian of C and write 0 for its identity element. Then
J is a g-dimensional abelian variety over Fq and it carries the “universal divisorial
correspondence with C.” More precisely, there is a divisorial correspondence M
between (C, x) and (J, 0) such that if S is another connected variety over Fq with
Fq -rational point s and L is a divisorial correspondence between (C, x) and (S, s),
then there is a unique morphism φ : S → J sending s to 0 such that L = φ∗ M.
(Of course M depends on the choice of base point x, but we omit this from the
notation.)
It follows that there is a canonical morphism, the Abel-Jacobi morphism, AJ :
C → J sending x to 0. Intuitively, this corresponds to the family of invertible
sheaves parameterized by C that sends y ∈ C to OC (y − x). More precisely, let
∆ ⊂ C × C be the diagonal, let
D = ∆ − x × C − C × x,
and let L = OC×C (D) which is a divisorial correspondence between (C, x) and itself.
The universal property above then yields the morphism AJ : C → J. It is known
that AJ is a closed immersion and that its image generates J as an algebraic group.
The second universal property enjoyed by J (or rather by AJ) is the Albanese
property: it is universal for maps to abelian varieties. More precisely, if A is an
abelian variety and φ : C → A is a morphism sending x to 0, then there is a unique
homomorphism of abelian varieties ψ : J → A such that φ = ψ ◦ AJ.
Combining the two universal properties gives a useful connection between corre-
spondences and homomorphisms: Suppose C and D are curves over Fq with rational
points x ∈ C and y ∈ D. Then we have an isomorphism
(5.1.1) DivCorr((C, x), (D, y)) ∼
= Hom(JC , JD ).
Intuitively, given a divisorial correspondence on C × D, we get a family of invertible
sheaves on D parameterized by C and thus a morphism C → JD . The Albanese
property then gives a homomorphism JC → JD . We leave the precise version as an
exercise, or see [Mil86b, 6.3]. We will use this isomorphism later to understand the
Néron-Severi group of a product of curves.
5.2. The Tate module
Let A be an abelian variety of dimension g over Fq , for example the Jacobian of
a curve of genus g. (See [Mil86a] for a brief introduction to abelian varieties and
[Mum08] for a much more complete treatment.) Choose a prime ` 6= p. Let A[`n ] be
the set of Fq points of A of order dividing `n . It is a group isomorphic to (Z/`n Z)2g
with a linear action of Gal(Fq /Fq ). We form the inverse limit
T` A = lim A[`n ]
←−
n
where the transition maps are given by multiplication by `. Let V` A = T` A ⊗Z` Q` ,
a 2g-dimensional Q` -vector space with a linear action of Gal(Fq /Fq ). It is often
called the Tate module of A.
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 221

According to Roquette, what we now call the Tate module seems to have first
been used in print by Deuring [Deu40] as a substitute for homology in his work on
correspondences on curves. It appears already in a letter of Hasse from 1935, see
[Roq06, p. 36].
The following proposition is the modern interpretation of the connection be-
tween homology and torsion points.
Proposition 5.2.1. Let A be an abelian variety over a field k and let ` be a prime
not equal to the characteristic of k. Let V` A be the Tate module of A and (V` A)∗
its dual as a Gk = Gal(k sep /k)-module.
• There is a canonical isomorphism of Gk -modules
(V` A)∗ ∼
= H 1 (A × k, Q` ).
• If A is the Jacobian of a curve C over k, then
H 1 (A × k, Q` ) ∼
= H 1 (C × k, Q` ).
For a proof of part 1, see [Mil86a, 15.1] and for part 2, see [Mil86b, 9.6].
Exercises 5.2.2. These exercises are meant to make the Proposition more plausi-
ble.
(1) Show that if A(C) is a complex torus Cg /Λ, then the singular homol-
ogy H1 (A(C), Q` ) is canonically isomorphic to V` A(C). (Hint: Use the
universal coefficient theorem to show that H1 (A(C), Z/`n Z) ∼
= Λ/`n Λ.)
(2) (Advanced) Let C be a smooth projective curve over an algebraically closed
field k. Let ` be a prime not equal to the characteristic of k. Use geo-
metric class field theory (as in [Ser88]) to show that unramified Galois
covers C 0 → C equipped with an isomorphism Gal(C 0 /C) ∼ = Z/`Z are in
bijection with elements of Hom(JC [`], Z/`Z). (Make a convention to deal
with the trivial homomorphism.) This suggests that H 1 (C, Z/`Z) “should
be” Hom(JC [`], Z/`Z) and H1 (C, Z/`Z) “should be” JC [`]. The reason we
only have “should be” rather than a theorem is that a non-trivial Galois
cover C 0 → C is never locally constant in the Zariski topology. This is a
prime motivation for introducing the étale topology.

6. Tate’s theorem on homomorphisms of abelian varieties

As usual, let k be a finite field and let A and B be two abelian varieties over k.
Choose a prime ` not equal to the characteristic of k and form the Tate modules
V` A and V` B. Any homomorphism of abelian varieties φ : A → B induces a ho-
momorphism of Tate modules φ∗ : V` A → V` B and this homomorphism commutes
with the action of Gk = Gal(k/k) on the Tate modules. We get an induced homo-
morphism Homk (A, B) ⊗ Q` → HomGk (V` A, V` B). Tate’s famous result [Tat66a]
asserts that this is an isomorphism:
Theorem 6.1. The map φ 7→ φ∗ induces an isomorphism of Q` -vector spaces:
Homk (A, B) ⊗ Q` →
˜ HomGk (V` A, V` B) .
We also mention [Zar08] which gives a different proof and a strengthening with
finite coefficients.
We will use Tate’s theorem in Theorem 12.1 of Lecture 2 to understand the
divisors on a product of curves in terms of homomorphisms between their Jacobians.
 LECTURE 1

Elliptic curves over function fields

In this lecture we discuss the basic facts about elliptic curves over function fields
over finite fields. We assume the reader has some familiarity with elliptic curves
over global fields such as Q or number fields, as explained, e.g., in [Sil09], and we
will focus on aspects specific to characteristic p. The lecture ends with statements
of the main results known about the conjecture of Birch and Swinnerton-Dyer in
this context.

1. Elliptic curves
1.1. Definitions
We write k = Fq for the finite field of cardinality q and characteristic p and we let
K be the function field of a smooth, projective, absolutely irreducible curve C over
k.
An elliptic curve over K is a smooth, projective, absolutely irreducible curve
of genus 1 over K equipped with a K-rational point O that will serve as the origin
of the group law.
All the basic geometric facts, e.g., of [Sil09, Ch. III and App. A], continue to
hold in the context of function fields. We review a few of them to establish notation,
but will not enter into full details.
Using the Riemann-Roch theorem, an elliptic curve E over K can always be
presented as a projective plane cubic curve defined by a Weierstrass equation, i.e.,
by an equation of the form
(1.1.1) Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3
where a1 , . . . , a6 ∈ K. The origin O is the point at infinity [0 : 1 : 0]. We often give
the equation in affine form:
(1.1.2) y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
where x = X/Z and y = Y /Z.
The quantities b2 , . . . , b8 , c4 , c6 , ∆, j are defined by the usual formulas ([Sil09,
III.1] or [Del75]). Since E is smooth, by the following exercise ∆ 6= 0.
Remark/Exercises 1.1.3. The word “smooth” in the definition of an elliptic
curve means that the morphism E → Spec K is smooth. Smoothness of a morphism
can be tested via the Jacobian criterion (see, e.g., [Har77, III.10.4] or [Liu02, 4.3.3]).
Show that the projective plane cubic (1.1.1) is smooth if and only if ∆ 6= 0.
Because the ground field K is not perfect, smoothness is strictly stronger than
the requirement that E be regular, i.e., that its local rings be regular local rings
(cf. [Liu02, 4.2.2]). For example, show that the projective cubic defined by Y 2 Z =
X 3 − tZ 3 over K = Fp (t) with p = 2 or 3 is a regular scheme, but is not smooth
over K.
223
224 LECTURE 1. ELLIPTIC CURVES OVER FUNCTION FIELDS

Definitions 1.1.4. Let E be an elliptic curve over K.

(1) We say E is constant if there is an elliptic curve E0 defined over k such
that E ∼ = E0 ×k K. Equivalently, E is constant if it can be defined by a
Weierstrass cubic (1.1.1) where the ai ∈ k.
(2) We say E is isotrivial if there exists a finite extension K 0 of K such that
E becomes constant over K 0 . Note that a constant curve is isotrivial.
(3) We say E is non-isotrivial if it is not isotrivial. We say E is non-constant
if it is not constant.
Remark/Exercises 1.1.5. Show that E is isotrivial if and only if j(E) ∈ k.
Suppose that E is isotrivial, so that E becomes constant over a finite extension
K 0 and let k 0 be the field of constants of K 0 (the algebraic closure of k in K 0 ). A
priori , the definition of isotrivial says that there is an elliptic curve E0 over k 0 such
that E ×K K 0 ∼ = E0 ×k0 K 0 . Show that we may take K 0 to have field of constants k
and E0 to be defined over k. Show also that we may take K 0 to be separable and
of degree dividing 24 over K.
Exercise 1.1.6. For any elliptic curve E over K, the functor on K-algebras L 7→
AutL (E × L) is represented by a group scheme Aut(E). (Concretely, this means
there is a group scheme Aut(E) such that for any K-algebra L, AutL (E × L) is
Aut(E)(L), the group of L-valued points of Aut(E).) Show that Aut(E) is an étale
group scheme. Equivalently, show that any element of AutK (E) is defined over a
separable extension of K. (This is closely related to the previous exercise.)
1.2. Examples
Let K = Fp (t) with p > 3 and define elliptic curves
E1 : y 2 = x3 + 1
E2 : y 2 = x3 + t6
E3 : y 2 = x3 + t
E4 : y 2 = x3 + x + t.
Then E1 ∼= E2 over K and both are constant, E3 is isotrivial and non-constant,
whereas E4 is non-isotrivial.
For more examples, let K = Fp (t) (with p restricted as indicated) and define
(p 6= 3) E5 : y 2 + ty = x3
(p 6= 2) E6 : y 2 = x3 + tx
(p arbitrary) E7 : y 2 + xy + ty = x3
(p arbitrary) E8 : y 2 + xy = x3 + tx
(p arbitrary) E9 : y 2 + xy = x3 + t.
Then E5 and E6 are isotrivial and non-constant whereas E7 , E8 , and E9 are non-
isotrivial.

2. Frobenius
If X is a scheme of characteristic p, we define the absolute Frobenius morphism
FrX : X → X as usual: It is the identity on the underlying topological space and
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 225

raises functions to the p-th power. When X = Spec K, FrX is just the map of
schemes induced by the ring homomorphism K → K, a 7→ ap .
Suppose as usual that K is a function field and let E be an elliptic curve over
K. Define a new elliptic curve E (p) over K by the fiber product diagram:
E (p) = Spec K ×Spec K E /E

 
Spec K
Fr / Spec K

More concretely, if E is presented as a Weierstrass cubic as in equation (1.1.2), then

E (p) is given by the equation with ai replaced by api . The universal property of the
fiber product gives a canonical morphism FrE/K , the relative Frobenius:
FrE/K
E FF / E (p) /E
FF
FF
FF
F"  
Spec K
Fr / Spec K

By definition FrE/K is a morphism over K. In terms of Weierstrass equations for

E and E (p) as above, it is just the map (x, y) 7→ (xp , y p ).
It is evident that FrE/K is an isogeny, i.e., a surjective homomorphism of elliptic
curves, and that its degree is p. We define V = VE/K to be the dual isogeny, so
that VE/K ◦ FrE/K = [p], multiplication by p on E.
Note that j(E (p) ) = j(E)p so that if E is non-isotrivial, E and E (p) are not
isomorphic. Thus, using Frobenius and its iterates, we see that there are infinitely
many non-isomorphic elliptic curves isogenous to any non-isotrivial E. This is in
marked contrast to the situation over number fields (cf. [Fal86]).
Lemma 2.1. Let E be an elliptic curve over K. Then j(E) is a p-th power in K
if and only if there exists an elliptic curve E 0 over K such that E ∼
= E 0(p) .
Proof. We sketch a fancy argument and pose as an exercise a more down-to-
earth proof. Obviously if there is an E 0 with E ∼ = E 0(p) , then j(E) = j(E 0(p) ) =
j(E ) ∈ K . Conversely, suppose j(E) ∈ K and choose an elliptic curve E 00
0 p p p

such that j(E 00 )p = j(E). It follows that E 00(p) is isomorphic to E over a finite
separable extension of K. In other words, E is the twist of E 00(p) by a cocycle in
H 1 (GK , AutK sep (E 00(p) )). But there is a canonical isomorphism AutK sep (E 00(p) ) ∼
=
AutK sep (E 00 ) and twisting E 00 by the corresponding element of
H 1 (GK , AutK sep (E 00 )) ∼
= H 1 (GK , AutK sep (E 00(p) ))
we obtain an elliptic curve E 0 with E 0(p) ∼ = E. 
Exercise 2.2. Use explicit equations, as in [Sil09, Appendix A], to prove the
lemma.

3. The Hasse invariant

Let F be a field of characteristic p and E an elliptic curve over F . Let OE be
the sheaf of regular functions on E and let Ω1E be the sheaf of Kähler differentials
on E. The coherent cohomology group H 1 (E, OE ) is a one-dimensional F -vector
226 LECTURE 1. ELLIPTIC CURVES OVER FUNCTION FIELDS

space and is Serre dual to the space of invariant differentials H 0 (E, Ω1E ). Choose a
non-zero differential ω ∈ H 0 (E, Ω1E ) and let η be the dual element of H 1 (E, OE ).
The absolute Frobenius FrE induces a (p-linear) homomorphism:
Fr∗E : H 1 (E, OE ) → H 1 (E, OE ).
We define an element A = A(E, ω) of F by requiring that Fr∗E (η) = A(E, ω)η. This
is the Hasse invariant of E. It has weight p − 1 in the sense that A(E, λ−1 ω) =
λp−1 A(E, ω) for all λ ∈ F × .
Suppose E is given by a Weierstrass equation (1.1.2) and ω = dx/(2y+a1 x+a3 ).
If p = 2, then A(E, ω) = a1 . If p > 2, choose an equation with a1 = a3 = 0. Then
A(E, ω) = the coefficient of xp−1 in (x3 + a2 x2 + a4 x + a6 )(p−1)/2 . These assertions
follow from [KM85, 12.4] where several other calculations of A are also presented.
Recall that E/K is ordinary if the group of p-torsion points E(K)[p] 6= 0
and supersingular otherwise. It is known that E is supersingular if and only if
A(E, ω) = 0 (e.g., [KM85, 12.3.6 and 12.4]) and in this case j(E) ∈ Fp2 (e.g.,
[KM85, proof of 2.9.4]). (Alternatively, one may apply [Sil09, V.3.1] to E over K.)
In particular, if E is supersingular, then it must also be isotrivial.

4. Endomorphisms
The classification of endomorphism rings in [Sil09, III.9] goes over verbatim to
the function field case: EndK (E) is either Z, an order in an imaginary quadratic
number field, or an order in a quaternion algebra over Q ramified exactly at ∞ and
p. The quaternionic case occurs if and only if E is supersingular, and the imaginary
quadratic case occurs if and only if j(E) is in Fp and E is not supersingular ([Sil09,
V.3.1 and Exer. V.5.8]).
In particular, if E is non-isotrivial, then EndK (E) = EndK (E) = Z.

5. The Mordell-Weil-Lang-Néron theorem

We write E(K) for the group of K-rational points of E and we call E(K) the
Mordell-Weil group of E over K. Lang and Néron (independently) generalized the
classical Mordell-Weil theorem to the function field context:
Theorem 5.1. Assume that K = Fq (C) is the function field of a curve over a
finite field and let E be an elliptic curve over K. Then E(K) is a finitely generated
abelian group.
(The theorems of Lang and Néron apply much more generally to any abelian
variety A over a field K that is finitely generated over its “constant field” k, but
one has to take care of the “constant part” of A. See [Ulm11] for details.)
We will not give a detailed proof of the MWLN theorem here, but will mention
two strategies. One is to follow the method of proof of the Mordell-Weil (MW)
theorem over a number field. Choose a prime number ` 6= p. By an argument very
similar to that in [Sil09, Ch. VIII] one can show that E(K)/`E(K) is finite (the
“weak Mordell-Weil theorem”) by embedding it in a Selmer group and showing
that the Selmer group is finite by using the two fundamental finiteness results of
algebraic number theory (finiteness of the class group and finite generation of the
unit group) applied to Dedekind domains in K. One can then introduce a theory of
heights exactly as in [Sil09] and show that the MW theorem follows from the weak
MW theorem and finiteness properties of heights. See the original paper of Lang
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 227

and Néron [LN59] for the full details. A complete treatment in modern language
has been given by Conrad [Con06].
One interesting twist in the function field setting comes if one takes ` = p above.
It is still true that the Selmer group for p is finite, but one needs to use the local
restrictions at all places; the maximal abelian extension of exponent p unramified
outside a finite but non-empty set of places is not finite and so one needs some
control on ramification at every place. See [Ulm91] for a detailed account of p-
descent in characteristic p.
A second strategy of proof, about which we will say more in Lecture 3, involves
relating the Mordell-Weil group of E to the Néron-Severi group of a closely related
surface E. In fact, finite generation of the Néron-Severi group (known as the “the-
orem of the base”) is equivalent to the Lang-Néron theorem. A direct proof of the
theorem of the base was given by Kleiman in [SGA6, XIII]. See also [Mil80, V.3.25].

6. The constant case

It is worth pausing in the general development to look at the case of a constant
curve E. Recall that K is the function field k(C) of the curve C over k = Fq .
Suppose E0 is an elliptic curve over k and let E = E0 ×k K.
Proposition 6.1. We have a canonical isomorphism
E(K) ∼
= Mork (C, E0 )
where Mork denotes morphisms of varieties over k (=morphisms of k-schemes).
Under this isomorphism, E(K)tor corresponds to the subset of constant morphisms.
Proof. By definition, E(K) is the set of K-morphisms
Spec K → E = E0 ×k K.
By the universal property of the fiber product, these are in bijection with k-
morphisms Spec K → E0 . Since C is a smooth curve, any k-morphism Spec K → E0
extends uniquely to a k-morphism C → E0 . This establishes a map E(K) →
Mork (C, E0 ). If η : Spec K → C denotes the canonical inclusion, composition with
η (φ 7→ φ ◦ η) induces a map Mork (C, E0 ) → E(K) inverse to the map above. This
establishes the desired bijection and this bijection is obviously compatible with the
group structures.
Since k is finite, it is clear that a constant morphism goes over to a torsion
point. Conversely, if P ∈ E(K) is torsion, say of order n, then the image of the
corresponding φ : C → E0 must lie in the set of n-torsion points of E0 , a discrete
set, and this implies that φ is constant. 

For example, if K is rational (i.e., C = P1 so that K = k(t)), then E(K) =

E0 (k).
Corollary 6.2. Let JC be the Jacobian of C. We have canonical isomorphisms
E(K)/E(K)tor ∼
= Homk−av (JC , E0 ) ∼
= Homk−av (E0 , JC ).
Proof. The Albanese property of the Jacobian of C (Subsection 5.1 of Lec-
ture 0) gives a surjective homomorphism
Mork (C, E0 ) → Homk−av (JC , E0 ).
228 LECTURE 1. ELLIPTIC CURVES OVER FUNCTION FIELDS

This homomorphism sends non-constant (and therefore surjective) morphisms to

non-constant (surjective) homomorphisms, so its kernel consists exactly of the con-
stant morphisms. The second isomorphism in the statement of the corollary follows
from the fact that Jacobians are self-dual. 
By Poincaré complete reducibility [Mil86a, 12.1], JC is isogenous to a product
of simple abelian varieties. Suppose JC is isogenous to E0m × A and A admits no
non-zero morphisms to E0 . We say that “E0 appears in JC with multiplicity m.”
Then it is clear from the corollary that E(K)/E(K)tor ∼ = Endk (E0 )m and so the
rank of E(K) is m, 2m, or 4m.
Tate and Shafarevich used these ideas to exhibit isotrivial elliptic curves over
F = Fp (t) of arbitrarily large rank. Indeed, using Tate’s theorem on isogenies
of abelian varieties over finite fields (reviewed in Section 6 of Lecture 0) and a
calculation of zeta functions in terms of Gauss sums, they were able to produce
a hyperelliptic curve C over Fp whose Jacobian is isogenous to E0m × A where E0
is a supersingular elliptic curve and the multiplicity m is as large as desired. If
K = Fp (C), E is the constant curve E = E0 × F , and E 0 is the twist of E by the
quadratic extension K/F , then Rank E 0 (F ) = Rank E(K) and so E 0 (F ) has large
rank by the analysis above. See the original article [TS67] for more details and a
series of articles by Elkies (starting with [Elk94]) for a beautiful application to the
construction of lattices with high packing densities.

7. Torsion
An immediate corollary of the MWLN theorem is that E(K)tor is finite. In fact,
E(K)tor is isomorphic to a group of the form
Z/mZ × Z/nZ
where m divides n and p does not divide m. (See for example [Sil09, Ch. 3].) One
can also see using the theory of modular curves that every such group appears for
a suitable K and E.
In another direction, one can give uniform bounds on torsion that depend only
on crude invariants of the field K.
Indeed, in the constant case, E(K)tor ∼ = E0 (Fq ) which has order bounded by
(q + 1)2 . In the isotrivial case, there is a finite extension K 0 with the same field
1/2

of constants k = Fq over which E becomes constant. Thus E(K)tor ⊂ E(L)tor

again has cardinality bounded by (q 1/2 + 1)2 .
We now turn to the non-isotrivial case.
Proposition 7.1. Assume that E is non-isotrivial and let gC be the genus of C.
Then there is a finite (and effectively calculable) list of groups—depending only on
gC and p—such that for any non-isotrivial elliptic curve E over K, E(K)tor appears
on the list.
Proof. (Sketch) First consider the prime-to-p torsion subgroup of E(K). It
has the form G = Z/mZ × Z/nZ where m|n and p 6 | m. There is a modu-
lar curve X(m, n), irreducible and defined over Fp (µm ), that is a coarse moduli
space for elliptic curves with subgroups isomorphic to G. We get a morphism
C → X(m, n) which is non-constant (because E is non-isotrivial) and therefore
surjective. The Riemann-Hurwitz formula then implies that gC ≥ gX(m,n) . But the
genus of X(m, n) goes to infinity with n. Indeed, gX(m,n) ≥ gX(1,n) and standard
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 229

genus formulae ([Miy06, 4.2]) together with crude estimation show that the latter
is bounded below by
n2 n log2 n
1+ − .
24ζ(2) 4
This shows that for a fixed value of gC , only finitely many groups G as above can
appear as E(K)tor .
The argument for p-torsion is similar, except that ones uses the Igusa curves
Ig(pn ) (cf. [KM85, Ch. 12]). If E(K) has a point of order pn , we get a non-
constant morphism C → Ig(pn ) and the genus of Ig(pn ) is asymptotic to p2n /48
[Igu68, p. 96]. 

This proposition seems to have been rediscovered repeatedly over the years.
The first reference I know of is [Lev68].
Since the genus of a function field is an analogue of the discriminant (more
precisely q 2g−2 is an analogue of the absolute value of the discriminant of a number
field), the proposition is an analogue of bounding E(K)tor in terms of the discrim-
inant of a number field K. One could ask for a strengthening where torsion is
bounded by “gonality”, i.e., by the smallest degree of a non-constant map C → P1 .
This would be an analogue of bounding E(K)tor in terms of the degree of a number
field K, as in the theorems of Mazur, Kamienny, and Merel [Mer96]. This is indeed
possible and can be proven by mimicking the proof of the proposition, replacing
bounds on the genus of the modular curve with bounds on its gonality. See [Poo07]
for the best results currently known on gonality of modular curves.
Exercise 7.2. Compute the optimal list mentioned in the proposition for g = 0.
(This is rather involved.) Note that the optimal list in fact depends on p. Indeed,
Z/11Z is on the list if and only if p = 11.
One can be very explicit about p-torsion:
Proposition 7.3. Suppose that E is a non-isotrivial elliptic curve over K. Then
E(K) has a point of order p if and only if j(E) ∈ K p and A(E, ω) is a (p − 1)st
power in K × .
Note that whether A(E, ω) is a (p − 1)st power is independent of the choice of
the differential ω.
Fr V
Proof. Let E −→ E (p) −→ E be the standard factorization of multiplication
by p into Frobenius and Verschiebung. Recall (e.g., [Ulm91, 2.1]) that A(E, ω) is a
(p − 1)st power in K if and only if ker Fr ∼ = µp if and only if ker V ∼
= Z/pZ if and
(p)
only if there is a non-trivial p-torsion point in E (K).
Now suppose that P ∈ E(K) is a non-trivial p-torsion point. Then Fr(P ) is a
non-trivial p-torsion point in E (p) (K) and so A(E, ω) is a (p − 1)st power in K. Let
E 0 be the quotient of E by the cyclic subgroup generated by P : E 0 = E/hP i. Since
hP i is in the kernel of multiplication by p, we have a factorization of multiplication
by p:
[p] : E → E 0 → E.
Since E → E 0 is étale of degree p and [p] is inseparable of degree p2 , we have that
E 0 → E is purely inseparable of degree p. But an elliptic curve in characteristic p
has a unique inseparable isogeny of degree p (namely the quotient by the unique
230 LECTURE 1. ELLIPTIC CURVES OVER FUNCTION FIELDS

connected subgroup of order p, the kernel of Frobenius) so we have an identification

E = E 0(p) . By 2.1, j(E) ∈ K p .
Conversely, suppose A(E, ω) is a (p − 1)st power and j(E) ∈ K p . Let E 0 be
the elliptic curve such that E 0(p) ∼
= E. Given a differential ω on E, there is a
differential ω 0 on E 0 such that A(E, ω) = A(E 0 , ω 0 )p (as can be seen for example
by using Weierstrass equations). It follows that A(E 0 , ω 0 ) is also a (p − 1)st power
in K. Thus we have a non-trivial point of order p in E 0(p) (K) = E(K). 

Part of the proposition generalizes trivially by iteration: if E(K) has a point

n
of order pn , then j(E) ∈ K p . A full characterization of pn torsion seems harder—
the condition that A(E, ω) be a (p − 1)st power is closely related to the equations
defining the Igusa curve Ig(p) ([KM85, 12.8]), but we do not have such explicit
equations for Ig(pn ) when n > 1.

8. Local invariants
Let E be an elliptic curve over K and let v be a place of K. A model (1.1.2) for E
with coefficients in the valuation ring O(v) is said to be integral at v. The valuation
of the discriminant ∆ of an integral model is a non-negative integer and so there
are models where this valuation takes its minimum value. Such models are minimal
integral models at v.
Choose a model for E that is minimal integral at v:
y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 .
Let ai ∈ κ(v) be the reductions of the coefficients and let Ev be the plane cubic
(8.1) y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
over the residue field κv . It is not hard to check using Weierstrass equations that
the isomorphism type of the reduced cubic (8.1) is independent of the choice of
minimal model.
If the discriminant of a minimal integral model at v has valuation zero, i.e.,
is a unit at v, then the reduced equation defines an elliptic curve over κv . If the
minimal valuation is positive, then the reduced curve is singular. We distinguish
three cases according to the geometry of the reduced curve.
Definition 8.2.
(1) If Ev is a smooth cubic, we say E has good reduction at v.
(2) If Ev is a nodal cubic, we say E has multiplicative reduction at v. If the
tangent lines at the node are rational over κ(v) we say the reduction is
split multiplicative and if they are rational only over a quadratic extension,
we say the reduction is non-split multiplicative.
(3) If Ev is a cuspidal cubic, we say E has additive reduction.
Define an integer av as follows:


 qv + 1 − #Ev (κv ) if E has good reduction at v

1 if E has split multiplicative reduction at v
(8.3) av =


 −1 if E has non-split multiplicative reduction at v
0 if E has additive reduction at v

 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 231

Exercise 8.4. To make this definition less ad hoc, note that in the good reduction
case, the numerator of the ζ-function of the reduced curve is 1 − av qv−s + qv1−2s .
Show that in the bad reduction cases, the ζ-function of the reduced curve is
1 − av qv−s
.
(1 − qv−s )(1 − qv1−s )
In the good reduction case, the results about zeta functions and étale cohomol-
√
ogy reviewed in Lecture 0, Sections 3 and 4 imply the “Hasse bound”: |av | ≤ 2 qv .
There are two more refined invariants in the bad reduction cases: the Néron
model and the conductor. The local exponent of the conductor at v, denoted nv is
defined as

0
 if E has good reduction at v
(8.5) nv = 1 if E has multiplicative reduction at v

2 + δv if E has additive reduction at v


Here δv is a non-negative integer that is 0 when p > 3 and is ≥ 0 when p = 2 or 3.

We refer to [Tat75] for a definition and an algorithm to compute δv . P
The (global) Pconductor of E is defined to be the divisor n = v nv [v]. Its
degree is deg n = v nv deg v.
The Néron model will be discussed in Lecture 3 below.
Exercise 8.6. Mimic [Sil09, Ch. VII] to define a filtration on the points of E over
a completion Kv of K. Show that the prime-to-p part of E(K)tor maps injectively
into E(Kv )/E(Kv )1 . Relate E(Kv )/E(Kv )1 to the special fiber of the Néron model
of E at v. As in the classical case, this gives an excellent way to bound the prime-
to-p part of E(K)tor .

9. The L-function
We define the L-function of E/K as an Euler product:
Y
−1 Y
−1
(9.1) L(E, T ) = 1 − av T deg v + qv T 2 deg v 1 − av T deg v
good v bad v

and
L(E, s) = L(E, q −s ).
(Here T is a formal indeterminant and s is a complex number. Unfortunately,
there is no standard reasonable parallel of the notations Z and ζ to distinguish the
function of T and the function of s.) Because of the Hasse bound on the size of
av , the product converges absolutely in the region Re s > 3/2, and as we will see
below, it has a meromorphic continuation to all s.
When E is constant it is elementary to calculate L(E, s) in terms of the zeta-
functions of E0 and C.
Exercise 9.2. Suppose that E = E0 ×k K. Write the ζ-functions of E0 and C as
rational functions: Q2 −s
i=1 (1 − αi q )
ζ(E0 , s) = −s 1−s
(1 − q )(1 − q )
and Q2gC −s
j=1 (1 − βj q )
ζ(C, s) = .
(1 − q −s )(1 − q 1−s )
232 LECTURE 1. ELLIPTIC CURVES OVER FUNCTION FIELDS

Prove that
− αi βj q −s )
Q
i,j (1
L(E, s) = Q2 Q2 .
−s ) 1−s )
i=1 (1 − αi q i=1 (1 − αi q

Thus L(E, s) is a rational function in q −s of degree 4gC − 4, it extends to a

meromorphic function of s, and it satisfies a functional equation for s ↔ 2 − s.
Its poles lie on the lines Re s = 1/2 and Re s = 3/2 and its zeroes lie on the line
Re s = 1.
Although the proofs are much less elementary, these facts extend to the non-
constant case as well:
Theorem 9.3. Suppose the E is a non-constant elliptic curve over K. Let n be the
conductor of E. Then L(E, s) is a polynomial in q −s of degree N = 4gC − 4 + deg n,
it satisfies a functional equation for s ↔ 2−s, and its zeroes lie on the line Re s = 1.
More precisely,
YN
L(E, s) = (1 − αi q −s )
i=1
where each αi is an algebraic integer of absolute value q in every complex embedding.
The collection of αi (with multiplicities) is invariant under αi 7→ q 2 /αi .
The theorem is a combination of results of Grothendieck, Deligne, and others.
We will sketch a proof of it in Lecture 4.
Note that in all cases L(E, s) is holomorphic at s = 1. In the non-constant
case, its order of vanishing at s = 1 is bounded above by N and it equals N if and
only if L(E, s) = (1 − q 1−s )N .

10. The basic BSD conjecture

This remarkable conjecture connects the analytic behavior of the function L(E, s),
constructed from local data, to the Mordell-Weil group, a global invariant.
Conjecture 10.1 (Birch and Swinnerton-Dyer).
Rank E(K) = ords=1 L(E, s)
The original conjecture was stated only for elliptic curves over Q [BSD65] but
it is easily seen to make sense for abelian varieties over global fields. There is
very strong evidence in favor of it, especially for elliptic curves over Q and abelian
varieties over function fields. See [Gro10, Lecture 3, §4] for a summary of the best
theoretical evidence in the number field case. We will discuss what is known for
elliptic curves in the function field case later in this course. See Section 12 for
statements of the main results and [Ulm11] for a discussion of the case of higher
dimensional abelian varieties over function fields.

11. The Tate-Shafarevich group

We define the Tate-Shafarevich group of E over K as
!
Y
(E/K) = ker H 1 (K, E) → H 1 (Kv , E) .
v
Here the cohomology groups can be taken to be Galois cohomology groups:
H 1 (K, E) = H 1 (GK , E(K sep ))
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 233

and similarly for H 1 (Kv , E); or they can be taken as étale or flat cohomology
groups of Spec K with coefficients in the sheaf associated to E. The flat cohomology
definition is essential for proving finer results on p-torsion in (E/K).
Exercise 11.1. Show that the group H 1 (K, E) (and therefore (E/K)) is torsion.
Hint: Show that given a class c ∈ H 1 (K, E), there is a finite Galois extension L/K
such that c vanishes in H 1 (L, E).
The refined BSD conjecture relates the leading coefficient of L(E, s) at s = 1 to
invariants of E including heights, Tamagawa numbers, and the order of (E/K).
In particular, the conjecture that (E/K) is finite is included in the refined BSD
conjecture. We will not discuss that conjecture in these lectures, so we refer to
[Gro10] and [Ulm11] for more details.

12. Statements of the main results

Much is known about the BSD conjecture over function fields. We start with general
results.
Theorem 12.1. Let E be an elliptic curve over a function field K. Then we have:
(1) Rank E(K) ≤ ords=1 L(E, s).
(2) The following are equivalent:
• Rank E(K) = ords=1 L(E, s)
• (E/K) is finite
• for any one prime number ` (` = p is allowed ), the `-primary part
(E/K)`∞ is finite.
(3) If K 0 /K is a finite extension and if the BSD conjecture holds for E over
K 0 , then it holds for E over K.
The theorem was proven by Tate [Tat66b] and Milne [Mil75] and we will sketch
a proof in Lecture 3. When the equivalent conditions of Item 2 hold, it turns out
that the refined BSD conjecture automatically follows. (This is also due to Tate
and Milne and will be discussed in detail in [Ulm11].)
We now state several special cases where the conjecture is known to be true.
As will be seen in the sequel, they all ultimately reduce either to Tate’s theorem
on isogenies of abelian varieties over finite fields (Theorem 6.1 of Lecture 0) or to
a theorem of Artin and Swinnerton-Dyer on K3 surfaces [ASD73].
Theorem 12.2. If E is an isotrivial elliptic curve over a function field K, then
the BSD conjecture holds for E.
Recall that a constant curve is also isotrivial.
To state the next result, we make an ad hoc definition. If E is an elliptic curve
over K = Fq (t) we define the height h of E to be the smallest non-negative integer
such that E can be defined by a Weierstrass equation (1.1.1) where the ai are all
polynomials and deg(ai ) ≤ hi. For example, the curves E1 and E2 in Subsection 1.2
have height h = 0 and the other curves E3 , . . . E9 there all have height h = 1. See
Section 4 of Lecture 3 below for a more general definition.
Theorem 12.3. Suppose that K = k(t) and that E is an elliptic curve over K of
height h ≤ 2. Then the BSD conjecture holds for E.
234 LECTURE 1. ELLIPTIC CURVES OVER FUNCTION FIELDS

Note that this case overlaps the preceding one since an elliptic curve over k(t)
is constant if and only if its height is zero (cf. Proposition 4.1 in Lecture 3).
The following case is essentially due to Shioda [Shi86]. To state it, consider a
polynomial f in three variables with coefficients in k which is the sum of exactly 4
non-zero monomials, say
4 3
e
X Y
f= ci xj ij
i=1 j=1
P3
where the ci ∈ k are non-zero. Set ei4 = 1 − j=1 eij and let A be the 4 × 4 integer
matrix A = (eij ). If det A 6= 0 (mod p), we say that f satisfies Shioda’s condition.
Note that the condition is independent of the order of the variables xj .
Theorem 12.4. Suppose that K = k(t) and that E is an elliptic curve over K.
Suppose that E is birational to a plane curve V (f ) ⊂ A2K where f is a polynomial
in k[t, x, y] ⊂ K[x, y] which is the sum of exactly 4 non-zero monomials and which
satisfies Shioda’s condition. Then the BSD conjecture holds for E.
For example, the theorem applies to the curves E4 , E7 , E8 , and E9 of Subsec-
tion 1.2 over K = Fq (t) for any prime power q. It applies more generally to these
curves when t is replaced by td for any d prime to p. Note that when d is large,
the height of the curve is also large, and so we get cases of BSD not covered by
Theorem 12.3.
Finally we state another more recent and ultimately much more flexible special
case due to Lisa Berger [Ber08].
Theorem 12.5. Suppose that K = k(t) and that E is an elliptic curve over K.
Suppose that E is birational to a plane curve of the form
f (x) = td g(y)
where f and g are rational functions of one variable and d is prime to p. Then the
BSD conjecture holds for E.
Here one should clear denominators to interpret the equation f = td g (or
work in a Zariski open subset of the plane). For example, if f (x) = x(x − 1) and
g(y) = y 2 /(1 − y) then we have the plane curve over K = k(t) defined by
x(x − 1)(1 − y) = td y 2
which turns out to be birational to
y 2 + xy + td y = x3 + td x2 .

13. The rest of the course

The remainder of these lectures will be devoted to sketching the proofs of most of
the main results and applying them to construct elliptic curves of large rank over
function fields.
More precisely, in Lecture 2 we will review facts about surfaces and the Tate
conjecture on divisors. This is a close relative of the BSD conjecture.
In Lecture 3 we will explain the relationship between the BSD and Tate con-
jectures and use it to prove the part of Theorem 12.1 related to ` 6= p as well as
most of the other theorems stated in the previous section.
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 235

In Lecture 4 we will recall a general result on vanishing of L-functions in towers

and combine it with the results above to obtain many elliptic curves of arbitrarily
large rank.
In Lecture 5, we will give other applications of these ideas to ranks of elliptic
curves and explicit points.
 LECTURE 2

Surfaces and the Tate conjecture

1. Motivation
Consider an elliptic curve E/K and suppose that K = k(t) and that we choose
an equation for E as in Lecture 1, equation (1.1.2) where the ai are in k[t]. Then
(1.1.2), viewed in K[x, y], defines an affine open subset of an elliptic curve E. But
if we view it as an equation in k[t, x, y], it defines an affine surface with a projection
to the affine t line. The generic fiber of this projection is the affine curve just
mentioned.
With a little more work (discussed in the next lecture), for any E over K = k(C)
we can define a smooth projective surface E over k with a morphism π : E → C whose
generic fiber is E. Obviously there will be close connection between the arithmetic
of E and that of E. Although E has higher dimension than E, it is defined over the
finite field k and as a result we have better control over its arithmetic. Pursuing
this line of inquiry leads to the main theorems stated at the end of the previous
section.
In this lecture, we discuss the relevant facts and conjectures about surfaces over
finite fields. In the next lecture we will look carefully at the connections between
E and E and deduce the main classical theorems.
There are many excellent references for the general theory of surfaces, including
[Bea96], [BHPV04], and [Băd01]. We generally refer to [Băd01] below since it works
throughout over a field of arbitrary characteristic.

2. Surfaces
Let k = Fq be a finite field of characteristic p. As always, by a surface over k
we mean a purely 2-dimensional, separated, reduced scheme of finite type over k.
Such a scheme is automatically quasi-projective and is projective if and only if it is
complete [Băd01, 1.28]. Since k is perfect, a surface X is a regular scheme if and
only if X → Spec k is a smooth morphism (e.g., [Liu02, 4.3.3, Exer. 3.24]). We
sloppily say that “X is smooth” if these conditions hold. Resolution of singularities
is known for surfaces: For any surface X , there is a proper birational morphism
X̃ → X with X̃ smooth. (We may even take this morphism to be a composition
of normalizations and blow ups at closed points [Lip78]. See also [Art86] for a nice
exposition.) Therefore, every surface is birational to a smooth projective surface. In
the cases of interest to us, this can be made very explicit in an elementary manner.
Throughout we assume that X is a smooth, projective, absolutely irreducible
surface over k and we assume that X (k) is non-empty, i.e., X has a k-rational point.
237
238 LECTURE 2. SURFACES AND THE TATE CONJECTURE

3. Divisors and the Néron-Severi group

We give a lightning review of divisors and equivalence relations on divisors. See,
for example, [Har77, V.1] for more details.
3.1. Divisor classes
A (Weil) divisor is a finite formal Z-linear combination of reduced, closed, codi-
mension 1 subvarieties of X : X
D= aZ Z.
In other words, the set of divisors is the free abelian group on the reduced, closed,
codimension 1 subvarieties on X .
If Z is a reduced, closed subvariety of X of codimension 1, there is an associated
valuation
ordZ : k(X )× → Z
that sends a rational function to its order of zero or pole along Z.
A rational function f on X has a divisor:
X
Div(f ) = ordZ (f )Z.
Z
A divisor D is said to be linearly equivalent to zero if there is a rational function
f such that Div(f ) = D. Two divisors D and D0 are linearly equivalent if their
difference D − D0 is linearly equivalent to zero.
The group of divisors modulo those linear equivalent to zero is the divisor class
group DivCl(X ). It is a fundamental invariant of X .
3.2. The Picard group
Let Pic(X ) be the Picard group of X , i.e., the group of isomorphism classes of
invertible sheaves on X with group law given by the tensor product. There is a
cohomological calculation of Pic(X ):
Pic(X ) ∼
= H 1 (X , O× ).
X
The map sending a divisor D to the invertible sheaf OX (D) induces an isomor-
phism DivCl(X )→
˜ Pic(X ).
3.3. The Néron-Severi group
As usual, we write X for X ×k k. We first introduce the notion of algebraic equiva-
lence for divisors on X . Intuitively, two divisors D and D0 are algebraically equiv-
alent if they lie in a family parameterized by a connected variety (which we may
take to be a smooth curve). More precisely, if T is a smooth curve over k and
D ⊂ X ×k T is a divisor that is flat over T , then we get a family of divisors on
X parameterized by T : t ∈ T corresponds to X × {t} ∩ D. Two divisors D1 and
D2 on X are algebraically equivalent if they lie in such a family, i.e., if there is
a curve T and a divisor D as above and two points t1 and t2 ∈ T (k) such that
Di = X ×k {ti } ∩ D. (A priori , to ensure transitivity of this relation we should
use chains of equivalences (see [Har77, Exer. V.1.7]) but see [Ful84, 10.3.2] for an
argument that shows the definition works as is.) Note that linear equivalence is
algebraic equivalence where T is restricted to be P1 ([Har77, Exer. V.1.7]) and so
algebraic equivalence is weaker than linear equivalence.
The group of divisors on X modulo those algebraically equivalent to zero is the
Néron-Severi group NS(X ). A classical (and difficult) theorem, the “theorem of
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 239

the base,” says that NS(X ) is finitely generated. See [LN59] and [SGA6, XIII.5.1]
for proofs and Lecture 3 below for more discussion. See also [Con06] for a modern
discussion of the results in [LN59].
Since linear equivalence is weaker than algebraic equivalence, NS(X ) is a quo-
tient of Pic(X ).
We define NS(X ) to be the image of Div(X ) in NS(X ) or equivalently the image
of Pic(X ) in NS(X ). Thus NS(X ) is again a finitely generated abelian group. As
we will see, it is of arithmetical nature.
Exercise 3.3.1. Let Gk = Gal(k/k). Show that NS(X ) is the group of invariants
NS(X )Gk . You will need to use that k is a finite field.

4. The Picard scheme

We define Pic0 (X ) as the kernel of the surjection Pic(X ) → NS(X ). In order
to understand this group better, we will introduce more structure on the Picard
group. The main fact we need to know is that the group Pic0 (X × k) is the set of
points on an abelian variety and is therefore a divisible group. (I.e., for every class
c ∈ Pic0 (X × k) and every positive integer n, there is a class c0 such that n c0 = c.)
Readers willing to accept this assertion can skip the rest of this section.
The Picard group Pic(X ) is the set of k-points of a group scheme. More pre-
cisely, under our hypotheses on X there is a group scheme called the Picard scheme
and denoted PicX /k which is locally of finite type over k and represents the relative
Picard functor. This means that if T → S = Spec k is a morphism of schemes and
πT : XT := X ×Spec k T → T is the base change then
Pic(XT )
PicX /k (T ) = .
πT∗ Pic(T )
Here the left hand side is the group of T -valued points of PicX /k . See [Kle05] for
a thorough and detailed overview of the Picard scheme, and in particular [Kle05,
9.4.8] for the proof that there is a scheme representing the relative Picard functor
as above.
We write Pic0X /k for the connected component of PicX /k containing the identity.
Under our hypotheses, Pic0X /k is a geometrically irreducible projective group scheme
over k [Kle05, 9.5.3, 9.5.4]. It may be non-reduced. (See examples in [Igu55] and
[Ser58]
 and
 a full analysis of this phenomenon in [Mum66].) We let PicVarX /k =
Pic0X /k , the Picard variety of X over k, which is an abelian variety over k.
red
0
If k is a field extension of k, we have
Pic0 (Xk0 ) = Pic0X /k (k 0 ) = PicVarX /k (k 0 )

so that Pic0 (Xk0 ) is the set of points of an abelian variety.

By [Kle05, 9.5.10], Pic0X /k (k) = Pic0 (X ), in other words, the class of a divisor
in Pic(X ) lies in Pic0 (X ) if and only if the divisor is algebraically equivalent to 0.

5. Intersection numbers and numerical equivalence

There is an intersection pairing on the Néron-Severi group:
NS(X ) × NS(X ) → Z
240 LECTURE 2. SURFACES AND THE TATE CONJECTURE

which is bilinear and symmetric. If D and D0 are divisors, we write D.D0 for their
intersection pairing.
There are two approaches to defining the pairing. In the first approach, one
shows that given two divisors, there are divisors in the same classes in NS(X ) (or
even the same classes in Pic(X )) that meet transversally. Then the intersection
number is literally the number of points of intersection. The work in this approach
is to prove a moving lemma and then show that the resulting pairing is well defined.
See [Har77, V.1] for the details.
In the second approach, one uses coherent cohomology. If L is an invertible
sheaf on X , let
X2
χ(L) = (−1)i dimk H i (X , L)
i=0
be the coherent Euler characteristic of L. Then define
D.D0 = χ(OX ) − χ(OX (−D)) − χ(OX (−D0 )) + χ(OX (−D − D0 )).
One checks that if C is a smooth irreducible curve on X , then C.D = deg OX (D)|C
and that if C and C 0 are two distinct irreducible curves on X meeting transversally,
then C.C 0 is the sum of local intersection multiplicities. See [Bea96, I.1-7] for
details. (Nowhere is it used in this part of [Bea96] that the ground field is C.)
Two divisors D and D0 are said to be numerically equivalent if D.D00 = D0 .D00
for all divisors D00 . If Num(X ) denotes the group of divisors in X up to numerical
equivalence, then we have surjections
Pic(X )  NS(X )  Num(X )
and so Num(X ) is a finitely generated group. It is clear from the definition that
Num(X ) is torsion-free and so we can insert NS(X )/tor (Néron-Severi modulo tor-
sion) into this chain:
Pic(X )  NS(X )  NS(X )/tor  Num(X ).

6. Cycle classes and homological equivalence

There is a general theory of cycle classes in `-adic cohomology, see for example
[SGA4 21 , [Cycle]]. In the case of divisors, things are much simpler and we can
construct a cycle class map from the Kummer sequence.
Indeed, consider the short exact sequence of sheaves on X for the étale topology:
`n
0 → µ`n → Gm −→ Gm → 0.
(The sheaves µ`n and Gm are perfectly reasonable sheaves in the Zariski topology
on X , but the arrow in the right is not surjective in that context. We need to use
the étale topology or a finer one.) Taking cohomology, we get a homomorphism
P ic(X )/`n = H 1 (X , Gm )/`n → H 2 (X , µ`n ).
Since Pic0 (X ) is a divisible group, we have NS(X )/`n = Pic(X )/`n and so taking
an inverse limit gives an injection
NS(X ) ⊗ Z` → H 2 (X , Z` (1)).
Composing with the natural homomorphism NS(X ) → NS(X ) gives our cycle
class map
(6.1) NS(X ) → NS(X ) ⊗ Z` → H 2 (X , Z` (1)).
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 241

We declare two divisors to be (`-)homologically equivalent if their classes in

H 2 (X , Z` (1)) are equal. (We will see below that this notion is independent of `.)
The group of divisors modulo homological equivalence will (temporarily) be denoted
Homol(X ). It will turn out to be a finitely generated free abelian group.
The intersection pairing on N S(X ) corresponds under the cycle class map to
the cup product on cohomology. This means that a divisor that is homologically
equivalent to zero is also numerically equivalent to zero. Thus we have a chain of
surjections:

Pic(X )  NS(X )  NS(X )/tor  Homol(X )  Num(X ).

7. Comparison of equivalence relations on divisors

A theorem of Matsusaka [Mat57] asserts that the surjection

NS(X )/tor  Num(X )

is in fact an isomorphism. Thus

NS(X )/tor ∼
= Homol(X ) ∼
= Num(X )
and these groups are finitely generated, free abelian groups. Since NS(X ) is finitely
generated, NS(X )tor is finite.
In all of the examples we will consider, NS(X ) is torsion free. (In fact, for
an elliptic surface with a section, the surjection NS(X ) → Num(X ) is always an
isomorphism, see [SS09, Theorem 6.5].) So to understand Pic(X ) we have only to
consider the finitely generated free abelian group NS(X ) and the group Pic0 (X ),
which is (the set of points of) an abelian variety.

Exercise 7.1. In the case of a surface X over the complex numbers, use the
cohomology of the exponential sequence
exp
×
0 → Z → OX −→ OX →0

to analyze the structure of Pic(X ).

8. Examples
8.1. P2
It is well known (e.g., [Har77, II.6.4]) that two curves on P2 are linearly equivalent
if and only if they have the same degree. It follows that Pic(P2 ) = NS(P2 ) ∼= Z.

8.2. P1 × P1
By [Har77, II.6.6.1], two curves on P1 × P1 are linearly equivalent if and only if they
have the same bi-degree. It follows that Pic(P1 × P1 ) = NS(P1 × P1 ) ∼ = Z2 .

8.3. Abelian varieties

If X is an abelian variety (of any dimension g), then Pic0 (X ) is the dual abelian
variety and NS(X ) is a finitely generated free abelian group of rank between 1 and
4g 2 . See [Mum08] for details.
242 LECTURE 2. SURFACES AND THE TATE CONJECTURE

8.4. Products of curves

Suppose that C and D are smooth projective curves over k with k-rational points
x ∈ C and y ∈ D. By definition (see Subsection 5.1 of Lecture 0), the group of
divisorial correspondences between (C, x) and (D, y) is a subgroup of Pic(C × D)
and it is clear that
Pic(C × D) ∼
= Pic(C) × Pic(D) × DivCorr ((C, x), (D, y))
∼
= Pic0 (C) × Pic0 (D) × Z2 × DivCorr ((C, x), (D, y)) .
Moreover, as we saw in Lecture 0,
DivCorr ((C, x), (D, y)) ∼
= Hom(JC , JD )
is a discrete group. It follows that
(8.4.1) Pic0 (C × D) ∼
= Pic0 (C) × Pic0 (D)
and
(8.4.2) NS(C × D) ∼
= Z2 × Hom(JC , JD ).
This last isomorphism will be important for a new approach to elliptic curves of
high rank over function fields discussed in Lecture 5.
8.5. Blow ups
Let X be a smooth projective surface over k and let π : Y → X be the blow up of
X at a closed point x ∈ X so that E = π −1 (x) is a rational curve on Y. Then we
have canonical isomorphisms
Pic(Y) ∼
= Pic(X ) ⊕ Z and NS(Y) ∼ = NS(X ) ⊕ Z
where in both groups the factor Z is generated by the class of E. See [Har77, V.3.2].
8.6. Fibrations
Let X be a smooth projective surface over k, C a smooth projective curve over k,
and π : X → C a non-constant morphism. Assume that the induced extension of
function fields k(C) ,→ k(X ) is separable and k(C) is algebraically closed in k(X ).
Then for every closed point y ∈ C, the fiber π −1 (y) is connected, and it is irreducible
for almost all y. Write F for the class in NS(X ) of the fiber over a k-rational point
y of C. (This exists because we assumed that X has a k-rational point.) We write
hF i for the subgroup of NS(X ) generated by F .
It is clear from the definition of NS(X ) that if y 0 is another closed point of C,
then the class in NS(X ) of π −1 (y 0 ) is equal to (deg y 0 )F .
Now suppose that z ∈ C is a closed point such that π −1 (z) is reducible, say
fz
X
π −1 (z) = ni Zi
i=1

where the Zi are the irreducible components of π −1 (z) and the ni are their mul-
tiplicities in the fiber. Then a consideration of intersection multiplicities (see for
example [Sil94, III.8]) shows that for any integers mi ,
X
mi Zi ∈ hF i ⊂ NS(X )
i
if and only if there is a rational number α such that mi = αni for all i. More
precisely, the intersection pairing restricted to the part of NS(X ) generated by the
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 243

classes of the Zi is negative semi-definite, with a one-dimensional kernel spanned

by integral divisors that are rational multiples of the whole fiber. It follows that
the subgroup of NS(X )/hF i generated by the classes of the Zi has rank fz − 1. It
is free of this rank if the gcd of the multiplicities ni is 1.
It also follows that if D is a divisor supported on a fiber of π and D0 is another
divisor supported on other fibers, then D = D0 in NS(X )/hF i if and only if D =
D0 = 0 in NS(X )/hF i.
Define L2 NS(X ) to be the subgroup of NS(X ) generated by all components of
all fibers of π over closed points of C. By the above, it is the direct sum of the
hF i and the subgroups of NS(X )/hF i generated by the components of the various
fibers. Thus we obtain the following computation of the rank of L2 NS(X ).
Proposition 8.6.1. For a closed point y of C, let fy denote the number of irre-
ducible components in the fiber π −1 (y). Then the rank of L2 NS(X ) is
X
1+ (fy − 1).
y

If for all y the greatest common divisor of the multiplicities of the components in
the fiber of π over y is 1, then L2 NS(X ) is torsion-free.

9. Tate’s conjectures T1 and T2

Tate’s conjecture T1 for X (which we denote T1 (X )) characterizes the image of the
cycle class map:
Conjecture 9.1 (T1 (X )). For any prime ` 6= p, the cycle class map induces an
isomorphism
NS(X ) ⊗ Q` → H 2 (X , Q` (1))Gk
We will see below that T1 (X ) is equivalent to the apparently stronger integral
statement that the cycle class induces an isomorphism
NS(X ) ⊗ Z` → H 2 (X , Z` (1))Gk
We will also see that T1 (X ) is independent of ` which is why we have omitted
` from the notation.
Since Gk is generated topologically by F rq , we have
H 2 (X , Q` (1))Gk = H 2 (X , Q` (1))F rq =1 = H 2 (X , Q` )F rq =q .
The injectivity of the cycle class map implies that
Rank NS(X ) ≤ dimQ` H 2 (X , Q` )F rq =q
and T1 (X ) is the statement that these two dimensions are equal.
The second Tate conjecture relates the zeta-function to divisors. Recall that
ζ(X , s) denotes the zeta function of X , defined in Lecture 0, Section 3.
Conjecture 9.2 (T2 (X )). We have
Rank NS(X ) = − ords=1 ζ(X , s)
Note that by the Riemann hypothesis, the poles of ζ(X , s) at s = 1 come from
P2 (X , q −s ). More precisely, using the cohomological formula (4.1) of Lecture 0 for
P2 , we have that the order of pole of ζ(X , s) at s = 1 is equal to the multiplicity of
q as an eigenvalue of F rq on H 2 (X , Q` ).
244 LECTURE 2. SURFACES AND THE TATE CONJECTURE

Thus we have a string of inequalities

(9.3) Rank NS(X ) ≤ dimQ` H 2 (X , Q` )F rq =q ≤ − ords=1 ζ(X , s).
Conjecture T1 (X ) is that the first inequality is an equality and conjecture T2 (X ) is
that the leftmost and rightmost integers are equal. It follows trivially that T2 (X )
implies T1 (X ). Tate proved the reverse implication.
Proposition 9.4. The conjectures T1 (X ) and T2 (X ) are equivalent. In particular,
T1 (X ) is independent of `.
Proof. First note that the intersection pairing on NS(X ) is non-degenerate,
so we get an isomorphism
NS(X ) ⊗ Q` =∼ Hom(NS(X ), Q` ).
On the other hand, the cup product on H 2 (X , Q` (1)) is also non-degenerate (by
Poincaré duality), so we have
H 2 (X , Q` (1)) ∼
= Hom(H 2 (X , Q` (1)), Q` ).
If we use a superscript Gk to denote invariants and a subscript Gk to denote coin-
variants, then we have a natural homomorphism
H 2 (X , Q` (1))Gk → H 2 (X , Q` (1))Gk
which is an isomorphism if and only if the subspace of H 2 (X , Q` (1)) where Frq acts
by 1 is equal to the whole of the generalized eigenspace for the eigenvalue 1. As we
have seen above, this holds if and only if we have
dimQ` H 2 (X , Q` )F rq =q = − ords=1 ζ(X , s).
Now consider the diagram
NS(X ) ⊗ Q` Hom(NS(X ), Q` )
O
h h∗

H 2 (X , Q` (1))Gk
f
/ H 2 (X , Q` (1))G Hom(H 2 (X , Q` (1))Gk , Q` ).
k

The lower right arrow is an isomorphism by elementary linear algebra. The maps
h and h∗ are the cycle map and its transpose and they are isomorphisms if and
only if T1 (X ) holds. One checks that the diagram commutes ([Tat66b, p. 24] or
[Mil75, Lemma 5.3]) and so T1 (X ) implies that f is an isomorphism. Thus T1 (X )
implies T2 (X ). 
We remark that the equality of dimQ` H 2 (X , Q` )F rq =q and − ords=1 ζ(X , s)
would follow from the semi-simplicity of F rq acting on H 2 (X , Q` ) (or even from its
semisimplicity on the F rq = q generalized eigenspace). This is a separate “stan-
dard” conjecture (see for example [Tat94]); it does not seem to imply T1 (X ).

10. T1 and the Brauer group

We define the (cohomological) Brauer group Br(X ) by
×
Br(X ) = H 2 (X , Gm ) = H 2 (X , OX )
(with respect to the étale or finer topologies). Because X is a smooth proper surface
over a finite field, the cohomological Brauer group is isomorphic to the usual Brauer
group (defined in terms of Azumaya algebras) and it is known to be a torsion group.
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 245

(See [Mil80, IV.2] and also three fascinating articles by Grothendieck collected in
[Gro68].) Artin and Tate conjectured in [Tat66b] that Br(X ) is finite.
Similarly, define
×
Br(X ) = H 2 (X , Gm ) = H 2 (X , OX ).
This group is torsion but need not be finite.
Taking the cohomology of the exact sequence
`n
0 → µ`n → Gm −→ Gm → 0
as in Section 6, we have an exact sequence
(10.1) 0 → NS(X )/`n → H 2 (X , µ`n ) → Br(X )`n → 0.
Taking Gk -invariants and then the inverse limit over powers of `, we obtain an exact
sequence
0 → NS(X ) ⊗ Z` → H 2 (X , Z` (1))Gk → T` Br(X ) → 0.
Since Br(X )` is finite, T` Br(X ) is zero if and only if the `-primary part of Br(X )
is finite. It follows that the ` part of the Brauer group is finite if and only if T1 (X )
for ` holds if and only if the integral version of T1 (X ) for ` holds. In particular,
since T1 (X ) is independent of `, if Br(X )[`∞ ] is finite for one `, then Br(X )[`∞ ] is
finite for all ` 6= p. It is even true, although more difficult to prove, that T1 (X ) is
equivalent to the finiteness of Br(X ).
Theorem 10.2. T1 (X ) holds if and only if Br(X ) is finite if and only if there is
an ` (` = p allowed ) such that the `-primary part of Br(X ) is finite.
Proof. We sketch the proof of the prime-to-p part of this assertion following
[Tat66b] and refer to [Mil75] for the full proof. We already noted that the `-primary
part of Br(X ) is finite for one ` 6= p if and only if T1 (X ) holds. To see that almost
all `-primary parts vanish, we consider the following diagram, which is an integral
version of the diagram in the proof of Proposition 9.4:
NS(X ) ⊗ Z`
e / Hom(NS(X ) ⊗ Z` , Z` ) Hom(NS(X ) ⊗ Q` /Z` , Q` /Z` )
O
h g∗

H 2 (X , Z` (1))Gk
f
/ H 2 (X , Z` (1))G Hom(H 2 (X , (Q` /Z` )(1))Gk , Q` /Z` )
k

Here e is induced by the intersection form, h is the cycle class map, f is induced
by the identity map of H 1 (X , Z` (1)) and g ∗ is the transpose of a map
g : NS(X ) ⊗ Q` /Z` → H 2 (X , (Q` /Z` )(1))
obtained by taking the direct limit over powers of ` of the first map in equa-
tion (10.1).
We say that a homomorphism φ : A → B of Z` -modules is a quasi-isomorphism
if it has a finite kernel and cokernel. In this case, we define
# ker(φ)
z(φ) = .
# coker(φ)
It is easy to check that if φ3 = φ2 φ1 (composition) and if two of the maps φ1 , φ2 ,
φ3 are quasi-isomorphisms, then so is the third and we have z(φ3 ) = z(φ2 )z(φ1 ).
In the diagram above, if we assume T1 (X ), then h is an isomorphism. The
map e is induced from the intersection pairing and is a quasi-isomorphism and
246 LECTURE 2. SURFACES AND THE TATE CONJECTURE

z(e) is (the ` part of) the order of the torsion subgroup of NS(X ) divided by (the
` part of) discriminant of the intersection form. We saw above that under the
assumption of T1 (X ), the map f is a quasi-isomorphism and it turns out that z(f )
is essentially (the ` part of) the leading term of the zeta function ζ(X , s) at s = 1.
In particular, under T1 (X ), e, f , and h are isomorphisms for almost all `. The same
must therefore be true of g ∗ . By taking Gk -invariants and a direct limit over powers
of ` in equation (10.1), one finds that z(g ∗ ) is equal to the order of Br(X )[`∞ ] and
so this group is trivial for almost all `. This completes our sketch of the proof of
the theorem. 

The sketch above has all the main ideas needed to prove that the prime-to-p part
of the Artin-Tate conjecture on the leading coefficient of the zeta function at s = 1
follows from the Tate conjecture T1 (X ). The p-part is formally similar although
more delicate. To handle it, Milne replaces the group in the lower right of the
diagram with the larger group Hom(H 2 (X , (Qp /Zp )(1)), Qp /Zp ). The z invariants
of the maps to and from this group turn out to have more p-adic content that
is related to the term q α (X ) in the Artin-Tate leading coefficient conjecture. We
refer to [Mil75] for the full details and to [Ulm11] for a discussion of several related
points, including the case p = 2 (excluded in Milne’s article, but now provable due
to improved p-adic cohomology) and higher dimensional abelian varieties.

11. The descent property of T1

If X̃ → X is the blow up of X at a closed point, then T1 (X̃ ) is equivalent to
T1 (X ). Indeed, under blowing up both the rank of NS(·) and the dimension of
H 2 (·, Q` (1))Gk increase by one. (See Example 8.5 above.) In fact:
Proposition 11.1. T1 (X ) is invariant under birational isomorphism. More gen-
erally, if X → Y is a dominant rational map, then T1 (X ) implies T1 (Y).
Proof. We give simple proof of the case where X and Y are surfaces. See
[Tat94] for the general case.
First, we may assume X 99KY is a morphism. Indeed, let X̃ → X be a blow up
resolving the indeterminacy of X 99KY, i.e., so that the composition X̃ → X 99KY
is a morphism. As we have seen above T1 (X ) implies T1 (X̃ ) so we may replace X
with X̃ and show that T1 (Y) holds.
So now suppose that π : X → Y is a dominant morphism. Since the dimensions
of X and Y are equal, π must be generically finite, say of degree d. But then the
push forward and pull-back maps on cycles present NS(Y) ⊗ Q` as a direct factor of
N S(X )⊗Q` ; they also present H 2 (Y, Q` (1)) as a direct factor of H 2 (X , Q` (1)). The
cycle class maps and Galois actions are compatible with these decompositions and
since by assumption N S(X ) ⊗ Q` →H ˜ 2 (X , Q` (1))Gk , we must also have N S(Y) ⊗
2 Gk
Q` →H
˜ (Y, Q` (1)) , i.e., T1 (Y). 

Note that the dominant rational map X 99KY could be a ground field extension,
or even a purely inseparable morphism.

12. Tate’s theorem on products

In this section we sketch how T1 for products of curves follows from Tate’s theorem
on endomorphisms of abelian varieties over finite fields.
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 247

Theorem 12.1 (Tate). Let C and D be curves over k and set X = C ×k D. Then
T1 (X ) holds.
Proof. Extending k if necessary, we may assume that C and D both have
rational points. Fix rational base points x and y (which we will mostly omit from
the notation below). Recall from Subsection 8.4 that
NS(C × D) ∼ = Z2 × DivCorr(C, D) ∼= Z2 × Hom(JC , JD ).
By the Künneth formula,
H 2 (X , Q` ) ∼
= H 2 (C, Q` ) ⊗ H 0 (D, Q` ) ⊕ H 0 (C, Q` ) ⊗ H 2 (D, Q` )

⊕ H 1 (C, Q` ) ⊗ H 1 (D, Q` )

∼
= Q` (−1) ⊕ Q` (−1) ⊕ H 1 (C, Q` ) ⊗ H 1 (D, Q` )

Twisting by Q` (1) and taking invariants, we have

Gk
H 2 (X , Q` (1))Gk = Q` 2 ⊕ H 1 (C, Q` ) ⊗ H 1 (D, Q` )(1) .
Under the cycle class map, the factor Z2 of NS(X ) (corresponding to C × {y} and
{x} × D) spans the factor Q` 2 of H 2 (X , Q` (1))Gk (corresponding to H 2 ⊗ H 0 and
H 0 ⊗ H 2 in the Kunneth decomposition). Thus what we have to show is that the
cycle class map induces an isomorphism

Gk
˜ H 1 (C, Q` ) ⊗ H 1 (D, Q` )(1)
Hom(JC , JD ) ⊗ Q` →
But H 1 (D, Q` )(1) ∼= H 1 (D, Q` )∗ ∼ = V` (JD ) and H 1 (C, Q` ) ∼
= V` (JC )∗ (∗ = Q` -
linear dual). Thus

G k
H 1 (C, Q` ) ⊗ H 1 (D, Q` )(1) ∼
= (V` (JC )∗ ⊗ V` (JD )) k ∼
G
= HomGk (V` (JC ), V` (JD )).
Thus the needed isomorphism is
Hom(JC , JD ) ⊗ Q` →
˜ HomGk (V` (JC ), V` (JD ))
and this is exactly the statement of Tate’s theorem (Lecture 0, Theorem 6.1). This
completes the proof of the theorem. 
Remarks 12.2.
(1) A variation of the argument above, using Picard and Albanese varieties,
shows that T1 for a product X × Y of varieties of any dimension follows
from T1 for the factors.
(2) It is worth noting that Tate’s conjecture T1 (and the proof of it for prod-
ucts of curves) only characterizes the image of in `-adic cohomology of
NS(X ) ⊗ Z` , not the image of NS(X ) itself. This should be contrasted
with the Lefschetz (1, 1) theorem, which characterizes the image of NS(X )
in deRham cohomology when the ground field is C.

13. Products of curves and DPC

Assembling the various parts of this lecture gives the main result:
Proposition 13.1. Let X be a smooth, projective surface over k. If there is a
dominant rational map
C ×k D99KX
from a product of curves to X , then the Tate conjectures T1 (X ) and T2 (X ) hold.
248 LECTURE 2. SURFACES AND THE TATE CONJECTURE

Indeed, by Theorem 12.1, we have T1 (C × D) and then by Proposition 11.1 we

deduce T1 (X ). By Proposition 9.4, T2 (X ) follows as well.
We say that “X is dominated by a product of curves (DPC).” The question of
which varieties are dominated by products of curves has been studied by Schoen
[Sch96]. In particular, over any field there are surfaces that are not dominated
by products of curves. Nevertheless, as we will see below, the collection of DPC
surfaces is sufficiently rich to give some striking results on the Birch and Swinnerton-
Dyer conjecture.
 LECTURE 3

Elliptic curves and elliptic surfaces

We keep our standard notations throughout this lecture: p is a prime, k = Fq

is the finite field of characteristic p with q elements, C is a smooth, projective,
absolutely irreducible curve over k, K = k(C) is the function field of C, and E is an
elliptic curve over K.

1. Curves and surfaces

In this section we will construct an elliptic surface E → C canonically associated to
an elliptic curve E/K. More precisely, we give a constructive proof of the following
result:
Proposition 1.1. Given an elliptic curve E/K, there exists a surface E over k
and a morphism π : E → C with the following properties: E is smooth, absolutely
irreducible, and projective over k, π is surjective and relatively minimal, and the
generic fiber of π is isomorphic to E. The surface E and the morphism π are
uniquely determined up to isomorphism by these requirements.
Here “the generic fiber of π” means EK , the fiber product:
EK := η ×C E /E

π
 
η = Spec K /C

“Relatively minimal” means that if E 0 is another smooth, absolutely irreducible,

projective surface over k with a surjective morphism π 0 : E 0 → C, then any birational
morphism E → E 0 commuting with π and π 0 is an isomorphism. Relative minimality
is equivalent to the condition that there are no rational curves of self-intersection
−1 in the fibers of π (i.e., to the non-existence of curves in fibers that can be blown
down).
Remarks 1.2. The requirements on E and π imply that π is flat and projective
and that all geometric fibers of π are connected. These properties of π will be
evident from the explicit construction below. It follows that π∗ OE ∼
= OC and more
generally that π is “cohomologically flat in dimension zero,” meaning that for every
morphism T → C the base change
πT : ET = E ×C T → T
satisfies πT ∗ OET = OT .
Uniqueness in Proposition 1.1 follows from general results on minimal models,
in particular [Lic68, Thm. 4.4]. See [Chi86] and [Liu02, 9.3] for other expositions.
249
250 LECTURE 3. ELLIPTIC CURVES AND ELLIPTIC SURFACES

We first give a detailed construction of a (possibly singular) “Weierstrass sur-

face” W → C and then resolve singularities to obtain E → C.
More precisely, the proposition follows from the following two results.
Proposition 1.3. Given an elliptic curve E/K, there exists a surface W over k
and a morphism π0 : W → C with the following properties: W is normal, absolutely
irreducible, and projective over k, π0 is surjective, each of its fibers is isomorphic
to an irreducible plane cubic, and its generic fiber is isomorphic to E.
This proposition is elementary, but does not seem to be explained in detail in
the literature, so we give a proof below.
Proposition 1.4. There is an explicit sequence of blow ups (along closed points
and curves in W) yielding a proper birational morphism σ : E → W where the
surface E and the composed morphism π = π0 ◦ σ : E → W → C have the properties
mentioned in Proposition 1.1.
Proof of Proposition 1.3. Choose a Weierstrass equation for E:
(1.5) y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6
where the ai are in K = k(C). Recall that we have defined the notion of a minimal
integral model at a place v of K: the ai should be integral at v and the valuation
at v of ∆ should be minimal subject to the integrality of the ai . Clearly, there is a
non-empty Zariski open subset U ⊂ C such that for every closed point v ∈ U , the
model (1.5) is a minimal integral model.
Let W1 be the closed subset of P2U := P2k ×k U defined by the vanishing of
Y 2 Z + a1 XY Z + a3 Y Z 2 − X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3


(1.6)
where X, Y, Z are the standard homogeneous coordinates on P2k . Then W1 is geo-
metrically irreducible and there is an obvious projection π1 : W1 → U (the restric-
tion to W1 of the projection P2U → U ). The fiber of π1 over a closed point v of U
is the plane cubic
Y 2 Z + a1 (v)XY Z + a3 (v)Y Z 2 = X 3 + a2 (v)X 2 Z + a4 (v)XZ 2 + a6 (v)Z 3
over the residue field κv at v. The generic point η of C lies in U and the fiber of π1
at η is E/K.
There are finitely many points in C \ U and we must extend the model W1 → U
over each of these points. Choose one of them, call it w, and choose a model of E
that is integral and minimal at w. In other words, choose a model of E
(1.7) y 02 + a01 x0 y 0 + a03 y 0 = x03 + a02 x02 + a04 x0 + a06
where the a0i ∈ K are integral at w and the valuation at w of the discriminant ∆ is
minimal. The new coordinates are related to the old by a transformation
(1.8) (x, y) = (u2 x0 + r, u3 y 0 + su2 x0 + t)
with u ∈ K × and r, s, t ∈ K. Let U 0 be a Zariski open subset of C containing w
on which all of the a0i are integral and the model (1.7) is minimal. Let W 0 be the
geometrically irreducible closed subset of P1U 0 defined by the vanishing of
Y 02 Z 0 + a01 X 0 Y 0 Z 0 + a03 Y 0 Z 02 − X 03 + a02 X 02 Z 0 + a04 X 0 Z 02 + a06 Z 0 3


 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 251

with its obvious projection π 0 : W 0 → U 0 . On the open set V = U ∩ U 0 , u is a unit

and the change of coordinates (1.8), or rather its projective version
(X, Y, Z) = (u2 X 0 + rZ, u3 Y 0 + su2 X 0 + tZ 0 , Z 0 )
defines an isomorphism between π1−1 (V ) and π 0−1 (V ) compatible with the projec-
tions. Glueing W1 and W 0 along this isomorphism yields a new surface W2 equipped
with a projection π2 : W2 → U2 where U2 = U ∪ U 0 . Note that U2 is strictly larger
than U . Moreover π2 is surjective, its geometric fibers are irreducible projective
plane cubics, and its generic fiber is E.
We now iterate this construction finitely many times to extend the original
model over all of C. We arrive at a surface W equipped with a proper, surjective
morphism π : W → C whose geometric fibers are irreducible plane cubics and whose
generic fiber is E. Since C is projective over k, so is W. Since W is obtained by
glueing reduced, geometrically irreducible surfaces along open subsets, it is also
reduced and geometrically irreducible. Since it has only isolated singular points,
by Serre’s criterion it is normal.
This completes the proof of Proposition 1.3. 
Note that the closure in W of the identity element of E is a divisor on W
which maps isomorphically to the base curve C. We write s0 : C → W for the
inverse morphism. This is the zero section of π0 . In terms of the coordinates on
W1 used in the proof above, it is just the map t 7→ ([0, 1, 0], t).
Discussion of Proposition 1.4. The algorithm mentioned in the Proposi-
tion is the subject of Tate’s famous paper [Tat75]. His article does not mention
blowing up, but the steps of the algorithm nevertheless give the recipe for the blow
ups needed. The actual process of blowing up is explained in detail in [Sil94, IV.9]
so we will not give the details here. Rather, we explain why there is a simple
algorithm, following [Con05].
First note that the surface W is reduced and irreducible and so has no embed-
ded components. Also, it has isolated singularities. (They are contained in the set
of singular points of fibers of π0 .) By Serre’s criterion, W is thus normal. Moreover,
and this is the key point, its singularities are rational double points. (See [Art86] for
the definition and basic properties of rational singularities and [Băd01, Chapters 3
and 4] for many more details. See [Con05, Section 8] for the fact that the singu-
larities of a minimal Weierstrass model are rational.) This implies that the blow
up of W at one of its singular points is again normal (so has isolated singularities)
and again has at worst rational double points. An algorithm to desingularize is
then simply to blow up at a singular point and iterate until the resulting surface is
smooth. Given the explicit nature of the equations defining W, finding the singular
points and carrying out the blow ups is straightforward.
In fact, Tate’s algorithm also calls for blowing up along certain curves. (This
happens at steps 6 and 7.) This has the effect of dealing with several singular points
at the same time, so is more efficient, but it is not essential to the success of the
algorithm.
This completes our discussion of Proposition 1.4. See below for a detailed
example covering a case not treated explicitly in [Sil94]. 
Conrad’s article [Con05] also gives a coordinate-free treatment of integral min-
imal models of elliptic curves.
252 LECTURE 3. ELLIPTIC CURVES AND ELLIPTIC SURFACES

It is worth remarking that Tate’s algorithm and the possible structures of the
bad fibers are essentially the same in characteristic p as in mixed characteristic.
On the other hand, for non-perfect residue fields k of characteristic p ≤ 3, there are
more possibilities for the bad fibers, in both equal and mixed characteristics—see
[Szy04].
The zero section of W lifts uniquely to a section which we again denote by
s0 : C → E.

2. The bundle ω and the height of E

We construct an invertible sheaf on C as follows, using the notation of the proof
of Proposition 1.1. Take the trivial invertible sheaf OU on U with its generating
section 1U . At each stage of the construction, extend this sheaf by glueing OU and
OU 0 over U ∩ U 0 by identifying 1U and u−1 1U 0 where u is the function appearing
in the change of coordinates (1.8).
The resulting invertible sheaf ω has several other descriptions. For example,
the sheaf of relative differentials Ω1E/C is invertible on the locus of E where π : E → C
is smooth (in particular in a neighborhood of the zero section) and, more or less
directly from the definition, ω can be identified with s∗0 (Ω1E/C ). Using relative
duality theory, ω can also be identified with the inverse of R1 π∗ OE . Finally, since
W as only rational singularities, ω is also isomorphic to R1 π0∗ OW
One may identify the coefficients ai of the Weierstrass equation locally defining
W with sections of ω i . Using this point of view, W can be identified with a closed
subvariety of a certain P2 -bundle over C. Namely, let V be the locally free OC
module of rank three
(2.1) V = ω 2 ⊕ ω 3 ⊕ OC
(where the exponents denote tensor powers). If PV denotes the projectivization of
V over C, a P2 bundle over C, then W is naturally the closed subset of PE defined
locally by the vanishing of Weierstrass equations as in (1.6).
Exercises 2.2. Verify the identifications and assertions in this section. In the case
where C = P1 , so K = k(t), check that ω = OP1 (h) where h is the smallest positive
integer such that E has a model (1.5) where the ai are in k[t] and deg ai ≤ hi.
Exercises 2.3. Check that c4 , c6 , and ∆ define canonical sections of ω 4 , ω 6 , and
ω 12 respectively, independent of the choice of equation for E. If p = 2 or 3, check
that b2 defines a canonical section of ω 2 and that c4 = b22 and c6 = −b32 . If p = 2,
check that a1 defines a canonical section of ω and that b2 = a21 . Note that since
positive powers of ω have non-zero sections, the degree of ω is non-negative.
Definition 2.4. The height of E, denoted h, is defined by h = deg(ω), the degree
of ω as an invertible sheaf on C.
Note that if E/K is constant (in the sense of Lecture 1) then the height of the
corresponding E is 0.

3. Examples
The case when C = P1 is particularly simple. First of all, one may choose a
model (1.5) that is integral and minimal simultaneously at every finite v, i.e., for
every v ∈ A1k . Indeed, start with any model and change coordinates so that the
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 253

ai are in k[t]. If w is a finite place where this model is not minimal, it is possible
(because k[t] is a PID) to choose a change of coordinates
(x, y) = (u2 x0 + r, u3 y 0 + su2 x0 + t)
where r, s, t, u ∈ k[t][1/w] and u a unit yielding a model that is minimal at w. Such
a change of coordinates does not change the minimality at any other finite place.
Thus after finitely many steps, we have a model integral and minimal at all finite
places. (This argument would apply for any K and any Dedekind domain R ⊂ K
which is a PID, yielding a model with the ai ∈ R that is minimal at all v ∈ Spec R.)
Focusing attention at t = ∞, there is a change of coordinates (1.8) with u = t−h
yielding a model integral and minimal at ∞. (Here h is minimal so that deg(ai ) ≤
hi.) So the bundle ω = O(h) = O(h∞).
As a very concrete example, consider the curve
y 2 = x(x + 1)(x + td )
over Fp (t) where p > 2 and d is not divisible by p. Since ∆ = 16t2d (td − 1)2 , this
model is integral and minimal at all non-zero finite places. It is also minimal at
zero as one may see by noting that c4 and c6 are units at 0. At infinity, the change
of coordinates
(x, y) = (t2h x0 , t3h y 0 )
with h = dd/2e yields a minimal integral model. Thus ω = O(h).
Working with Tate’s algorithm shows that E has I2 reduction at the d-th roots
∗
of unity, I2d reduction at t = 0, and either I2d or I2d reduction at infinity depending
on whether d is odd or even.
Since the case of In reduction is not treated explicitly in [Sil94], we give more
details on the blow ups needed to resolve the singularity over t = 0. In terms of the
coordinates on W1 used in the proof of Proposition 1.4 we can consider the affine
surface defined by
x3 + (td + 1)x2 + td x − y 2 = 0
which is an open neighborhood of the singularity at x = y = t = 0. If d = 1, then
the tangent cone is the irreducible plane conic defined by x2 + tx − y 2 = 0. The
singular point thus blows up into a smooth rational curve and it is easy to check
that the resulting surface is smooth in a neighborhood of the fiber t = 0. Now
assume that d > 1. Then the tangent cone is the reducible conic x2 − y 2 = 0 and
so the singular point blows up into two rational curves meeting at one point. More
precisely, the blow up is covered by three affine patches. In one of them, the surface
upstairs is
tx31 + (td + 1)x21 + td−1 x1 − y12 = 0
and the morphism is x = tx1 , y = ty1 . The exceptional divisor is the reducible curve
t = x21 − y12 = 0 and the point of intersection of the components t = x1 = y1 = 0 is
again a double point. Considering the other charts shows that there are no other
singular points in a neighborhood of t = 0 and that the exceptional divisor meets
the original fiber over t = 0 in two points. We now iterate this process d − 1 times,
introducing two new components at each stage. After d−1 blow ups, the interesting
part of our surface is given by
td−1 x3d−1 + (td + 1)x2d−1 + txd−1 − yd−1
2
= 0.
254 LECTURE 3. ELLIPTIC CURVES AND ELLIPTIC SURFACES

At this last stage, blowing up introduces one more component meeting the two
components introduced in the preceding step at one point each. The (interesting
part of the) surface is now
td x3d + (td + 1)x2d + xd − yd2 = 0
which is regular in a neighborhood of t = 0. Thus we see that the fiber over t = 0
in E is a chain of 2d rational curves, i.e., a fiber of type I2d .
The resolution of the singularities over points with td = 1 is similar but simpler
because only one blow up is required. At t = ∞, if d is even then the situation is
very similar to that over t = 0 and the reduction is again of type I2d . If d is odd,
∗
the reduction is of type I2d . We omit the details in this case since it is treated fully
in [Sil94].
Exercise 3.1. In the table in Tate’s algorithm paper [Tat75] (and the slightly more
precise version in [Sil09, p. 448]), the last three rows have restrictions on p. Give
examples showing that these restrictions are all necessary for the discriminant and
conductor statements, and for the statement about j in the In∗ , p = 2 case. Show
that the other assertions about the j-invariant are correct for all p.

4. E and the classification of surfaces

It is sometimes useful to know how E fits into the Enriques-Kodaira classification
of surfaces. In this section only, we replace k with k and write E for what elsewhere
is denoted E.
Recall that the height of E is defined as h = deg ω.
Proposition 4.1. ω ∼
= OC if and only if E is constant. If h = deg(ω) = 0, then E
is isotrivial.
Proof. It is obvious that if E is constant, then ω ∼= OC . Conversely, suppose
∼
ω = OC . Then the construction of π0 : W → C in Proposition 1.3 yields an
irreducible closed subset of P2C (because the P2 -bundle PV in (2.1) is trivial):
W ⊂ P2C = P2k ×k C.
Let σ : W → P2k be the restriction of the projection P2C → P2k . Then σ is not
surjective (since most points in the line at infinity Z = 0 are not in the image) and
so its image has dimension < 2. Considering the restriction of σ to a fiber of π0
shows that the image of σ is in fact an elliptic curve E0 and then it is obvious from
dimension considerations that
W = π0−1 (π0 (W)) = E0 × C.
It follows that E, the generic fiber of π0 , is isomorphic to E0 × Spec K, i.e., that E
is constant.
Now assume that h = 0. Then ∆ is a non-zero global section of the invertible
sheaf ω 12 on C of degree 0. Thus ω 12 is trivial. It follows that there is a finite
unramified cover of C over which ω becomes trivial and so by the first part, E
becomes constant over a finite extension, i.e., E is isotrivial. 
Note that E being isotrivial does not imply that h = 0.
Exercise 4.2. Give an example of a non-constant E of height zero. Hint: Consider
the quotient of a product of elliptic curves by a suitable free action of a group of
order two.
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 255

Proposition 4.3. The canonical bundle of E is Ω2E ∼

= π ∗ Ω1C ⊗ ω .

Here we are using that E → C has a section and therefore no multiple fibers.
The proof, which we omit, proceeds by considering R1 π∗ OE and using relative
duality. See for example [Băd01, 7.15].
We now consider several cases:
If 2gC − 2 + h > 0, then it follows from the Proposition that the dimension of
n
H 0 (E, (Ω2 )⊗ ) grows linearly with n, so E has Kodaira dimension 1.
If 2gC − 2 + h = 0, then the Kodaira dimension of E is zero and there are two
possibilities: (1) gC = 1 and h = 0; or (2) gC = 0 and h = 2. In the first case,
there is an unramified cover of C over which E becomes constant and so E is the
quotient of a product of two elliptic curves. These surfaces are sometimes called
“bi-elliptic.” In the second case, Ω2E = OE and H 1 (E, OE ) = H 0 (C, ω −1 ) = 0 and
so E is a K3 surface.
If 2gC − 2 + h < 0, then the Kodaira dimension of E is −∞ and there are again
two possibilities: (1) gC = 0 and h = 1, in which case E is a rational surface by
Castelnuovo’s criterion; or (2) gC = 0 and h = 0, in which case E is constant and
E is a ruled surface E0 × C = E0 × P1 .

5. Points and divisors, Shioda-Tate

If D is an irreducible curve on E, then its generic fiber
D.E := D ×C E
is either empty or is a closed point of E. The former occurs if and only if D is
supported in a fiber of π. In the latter case, the residue degree of D.E is equal to
the generic degree of D → C. Extending by linearity, we get homomorphism
Div(E) → Div(E)
whose kernel consists of divisors supported in the fibers of π.
There is a set-theoretic splitting of this homomorphism, induced by the map
sending a closed point of E to its scheme-theoretic closure in E. However, this is
not in general a group homomorphism.
Let L1 Div(E) be the subgroup of divisors D such that the degree of D.E is
zero and let L2 Div(E) be subgroup such that D.E = 0. We write Li Pic(E) and
Li NS(E) (i = 1, 2) for the images of Li (E) in Pic(E) and NS(E) respectively.
The Shioda-Tate theorem relates the Néron-Severi group of E to the Mordell-
Weil group of E:
Theorem 5.1. If E → C is non-constant, D 7→ D.E induces an isomorphism
L1 NS(E) ∼
= E(K)
L2 NS(E)
If E → C is constant, we have
L1 NS(E) ∼
= E(K)/E(k)
L2 NS(E)
This theorem seems to have been known to the ancients (Lang, Néron, Weil, ...)
and was stated explicitly in [Tat66b] and in papers of Shioda. A detailed proof in
a more general context is given in [Shi99]. Note however that in [Shi99] the ground
field is assumed to be algebraically closed. See [Ulm11] for the small modifications
needed to treat finite k.
256 LECTURE 3. ELLIPTIC CURVES AND ELLIPTIC SURFACES

It is obvious that N S(E)/L1 N S(E) is infinite cyclic.

P We saw in Example 8.6
of Lecture 2 that L2 N S(E) is free abelian of rank 1 + v (fv − 1). So as a corollary
of the theorem, we have the following rank formula, known as the Shioda-Tate
formula:
X
(5.2) Rank E(K) = Rank N S(E) − 2 − (fv − 1)
v

For more on the geometry of elliptic surfaces and elliptic curves over function
fields, with an emphasis on rational and K3 surfaces, I recommend [SS09].

6. L-functions and Zeta-functions

We are going to relate the L-function of E and the zeta function of E. We note that
from the definition, Z(E, T ) depends only on the underlying set of closed points of
E and we may partition this set using the map π.
We have
Y  −1
Z(E, T ) = 1 − T deg(x)
closed x∈E
Y Y  −1
= 1 − T deg(x)
closed y∈C x∈π −1 (y)
Y
= Z(π −1 (y), T deg(y) )
closed y∈C

For y such that π −1 (y) is a smooth elliptic curve, we know that

(1 − ay T + qy T 2 )
Z(π −1 (y), T ) =
(1 − T )(1 − qy T )

and the numerator here is the factor that enters into the definition of L(E, T ).
To complete the calculation, we need an analysis of the contribution of the bad
fibers. We consider the fiber π −1 (y) as a scheme of finite type over the residue field
κy , the field of qy elements. As such, it has irreducible components. Its “geometric
components” are the components of the base change to κy ; these are defined over
some finite extension of κy .
For certain reduction types (In , In∗ (n ≥ 0), IV and IV ∗ ) it may happen that all
the geometric components are defined over κy , in which case we say the reduction
is “split”, or it may happen that some geometric components are only defined over
a quadratic extension of κy , in which case we say the reduction is “non-split.” This
agrees with the standard usage in the case of In reduction and may be non-standard
in the other cases.

Proposition 6.1. The zeta function of the a singular fiber of π has the form

(1 − T )a (1 + T )b
Z(π −1 (y), T ) =
(1 − qy T )f (1 + qy T )g
1 (1 − T )a+1 (1 + T )b
=
(1 − T )(1 − qy T ) (1 − qy T )f −1 (1 + qy T )g
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 257

where the integers a, b, f , and g are determined by the reduction type at y and are
given in the following table:
a b f g
split In 0 0 n 0
non-split In , n odd −1 1 (n + 1)/2 (n − 1)/2
non-split In , n even −1 1 n/2 + 1 (n − 2)/2
split In∗ −1 0 5+n 0
non-split In∗ −1 0 4+n 1
II −1 0 1 0
II ∗ −1 0 9 0
III −1 0 2 0
III ∗ −1 0 8 0
split IV −1 0 3 0
non-split IV −1 0 2 1
split IV ∗ −1 0 7 0
non-split IV ∗ −1 0 3 4

Exercise 6.2. Use an elementary point-counting argument to verify the propo-

sition. In particular, check that the number of components of π −1 (y) that are
rational over κy is f and that the order of pole at T = qy−1 of
Z(π −1 (y), T )(1 − T )(1 − qy T )
is f − 1.
Using the Proposition and the definition of the L-function (in Lecture 1, equa-
tion (9.1)) we find that
Z(C, T )Z(C, qT ) Y (1 − T )av +1 (1 + T )bv
(6.3) L(E, T ) =
Z(E, T )
bad v
(1 − qv T deg(v) )fv −1 (1 + qv T deg(v) )gv
where av , bv , fv and gv are the invariants defined in the Proposition at the place
v. Using the Weil conjectures (see Section 3 of Lecture 0), we see that the orders
of L(E, s) and ζ(E, s) at s = 1 are related as follows:
X
(6.4) ords=1 L(E, s) = − ords=1 ζ(E, s) − 2 − (fv − 1).
v

Remark 6.5. This simple approach to evaluating the order of zero of the L-function
does not yield the important fact that L(E, T ) is a polynomial in T when E is non-
constant, nor does it yield the Riemann hypothesis for L(E, T ).
For a slightly more sophisticated (and less explicit) comparison of ζ-functions
and L-functions in a more general context, see [Gor79].

7. The Tate-Shafarevich and Brauer groups

The last relationship between E and E we need concerns the Tate-Shafarevich and
Brauer groups.
Theorem 7.1. Suppose that E is an elliptic curve over K = k(C) and E → C
is the associated elliptic surface as in Proposition 1.1. Then there is a canonical
isomorphism
Br(E) ∼
= (E/K).
258 LECTURE 3. ELLIPTIC CURVES AND ELLIPTIC SURFACES

The proof of this result, which is somewhat involved, is given in [Gro68, Sec-
tion 4]. The main idea is simple enough: one computes Br(E) = H 2 (E, Gm ) using
the morphism π : E → C and a spectral sequence. Using that the Brauer group of
a smooth, complete curve over a finite field vanishes, one finds that the main term
is H 1 (C, R1 π∗ Gm ). Since R1 π∗ Gm is the sheaf associated to the relative Picard
group, it is closely related to the sheaf on C represented by the Néron model of
E. This provides a connection with the Tate-Shafarevich group which leads to the
theorem.
See [Ulm11] for more details about this and the closely related connection
between H 2 (E, Z` (1))Gk and the `-Selmer group of E.

8. The main classical results

We are now in a position to prove the theorems of Section 12 of Lecture 1. For
convenience, we restate Theorem 12.1 and a related result.
Theorem 8.1. Suppose that E is an elliptic curve over K = k(C) and E → C is
the associated elliptic surface as in Proposition 1.1.
(1) BSD holds for E if and only if T2 holds for E.
(2) Rank E(K) ≤ ords=1 L(E, s).
(3) The following are equivalent:
• Rank E(K) = ords=1 L(E, s)
• (E/K) is finite
• for any one prime number ` (` = p is allowed ), the `-primary part
(E/K)`∞ is finite.
(4) If K 0 /K is a finite extension and if the BSD conjecture holds for E over
K 0 , then it holds for E over K.
Proof. Comparing (5.2) and (6.4), we have that
Rank E(K) − ords=1 L(E, s) = Rank N S(E) + ords=1 ζ(E, s).
Since BSD is the assertion that the left hand side is zero and T2 is the assertion
that the right hand side is zero, these conjectures are equivalent.
By Theorem 9.3 of Lecture 2, the right hand side is ≤ 0 and therefore so is the
left. This gives the inequality Rank E(K) ≤ ords=1 L(E, s).
The statements about (E/K) follow from Theorem 7.1 ( (E/K) ∼ = Br(E)),
the equivalence of BSD and T2 (E), and Theorem 10.2 of Lecture 2.
The last point follows from the equivalence of BSD and T2 (E) and Proposi-
tion 11.1 of Lecture 2. 

Proofs of Theorems 12.2 and 12.3 of Lecture 1. Theorem 12.2 of Lec-

ture 1 concerns isotrivial elliptic curves. By the last point of Theorem 8.1 above,
it suffices to show that BSD holds for constant curves. But if E is constant, then
E is a product of curves, so the Tate conjecture for E follows from Theorem 12.1 of
Lecture 2. The first point of Theorem 8.1 above then gives BSD for E.
Theorem 12.3 of Lecture 1 concerns elliptic curves over k(t) of low height. By
the discussion in Section 4, if E/k(t) has height ≤ 2 then E is a rational or K3
surface. (Strictly speaking, this is true only over a finite extension of k, but the last
point of Theorem 8.1 allows us to make this extension without loss of generality.)
But T2 (X ) for a rational surface follows from Proposition 13.1 of Lecture 2. For E
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 259

such that E is a K3 surfaces, Artin and Swinnerton-Dyer proved the finiteness of

(E/K) (and therefore BSD) in [ASD73]. 

9. Domination by a product of curves

Combining part 1 of Theorem 8.1 with Proposition 13.1 of Lecture 2, we have the
following.
Theorem 9.1. Let E be an elliptic curve over K with associated surface E. If E
is dominated by a product of curves, then BSD holds for E.
Theorem 12.4 (“four monomials”) and Berger’s theorem 11.1 are both corol-
laries of Theorem 9.1, as we will explain in the remainder of this lecture.

10. Four monomials

We recall Shioda’s conditions. Suppose that f ∈ R = k[x1 , x2 , x3 ] is the sum of
exactly four non-zero monomials:
4 3
e
X Y
f= ci xj ij
i=1 j=1
P3
where ci ∈ k and the eij are non-negative integers. Let ei4 = 1 − j=1 eij and form
the 4 × 4 matrix A = (eij ). Assuming that det(A) 6= 0 (in Z), let δ be the smallest
positive integer such that there is a 4 × 4 integer matrix B with AB = δI4×4 . We
say that f satisfies Shioda’s 4-monomial condition if δ 6= 0 in k, i.e., if p 6 | δ. The
following exercise shows that this is equivalent to the definition in Lecture 1.
Exercises 10.1. Show that a prime ` divides δ if and only if it divides det(A).
P3
Show that if we change the definition of ei4 to ei4 = d − j=1 eij for some other
non-zero integer d and define δd using the new A = (eij ), then δ1 divides δd for all
d. I.e., d = 1 is the optimal choice to minimize δ.
Exercise 10.2. With ci and eij as above, show that the system of equations
4
e
Y
dj ij = c−1
i i = 1, . . . , 4
j=1

has a solution with dj ∈ Fq , j = 1, . . . , 4.

Proof of Theorem 12.4 of Lecture 1. Briefly, the hypotheses imply that
the associated elliptic surface E → P1 is dominated by a Fermat surface (of degree
δ) and thus by a product of Fermat curves (of degree δ). Thus Theorem 9.1 implies
that BSD holds for E.
In more detail, note that E is birational to the affine surface V (f ) ⊂ A3k . So it
will suffice to show that V (f ) is dominated by a product of curves. To that end,
it will be convenient to identify k[t, x, y] and R = k[x1 , x2 , x3 ] by sending t 7→ x1 ,
x 7→ x2 and y 7→ x3 , so that f becomes
4 3
e
X Y
f= ci xj ij .
i=1 j=1
260 LECTURE 3. ELLIPTIC CURVES AND ELLIPTIC SURFACES

Exercise 10.2 implies that, after extending k if necessary, we may change coor-
dinates (xj 7→ dj xj ) so that the coefficients ci are all 1. Then the matrix A defines
rational a map φ from V (f ) to the Fermat surface of degree 1
F12 = {y1 + y2 + y3 + y4 = 0} ⊂ P3k ,
Q4 e
namely φ∗ (yi ) = j=1 xj ij . Similarly, the matrix B defines a rational map ψ from
the Fermat surface of degree δ
Fδ2 = {z1δ + z2δ + z3δ + z4δ = 0} ⊂ P3k
Q4 B
to V (f ), namely ψ ∗ (xi ) = j=1 zj ij . The composition of these maps is the stan-
dard projection from Fδ2 to F12 , namely yi 7→ ziδ and so both maps are dominant.
Finally, Shioda and Katsura [SK79] showed that Fδ2 is dominated by the prod-
uct of Fermat curves Fδ1 ×Fδ1 . Thus, after extending k, E is dominated by a product
of curves and Theorem 9.1 finishes the proof. 
As we will explain below, this Theorem can be combined with results on an-
alytic ranks to give examples of elliptic curves over Fp (t) with arbitrarily large
Mordell-Weil rank. (In fact, similar ideas can be used to produce Jacobians of
every dimension with large rank. For this, see [Ulm07] and also [Ulm11].)
Unfortunately, Theorem 12.4 is very rigid—as one sees in the proof, varying
the coefficients in the 4-nomial f does not vary the isomorphism class of E over Fq
and so we get only finitely many non-isomorphic elliptic curves over Fp (t). Berger’s
construction, explained in the next subsection, was motivated by a desire to over-
come this rigidity and give families of examples of curves where one knows the BSD
conjecture.

11. Berger’s construction

Berger gave a much more flexible construction of surfaces that are dominated by a
product of curves in a tower. More precisely, we note that if E → P1 is an elliptic
surface and φ : P1 → P1 is the morphism with φ∗ (t) = ud (corresponding to the
field extension k(u)/k(t) with ud = t), then it is not in general the case that the
base changed surface
E 0 = E × P1 / P1
k k

 
P1k
φ
/ P1
k

is dominated by a product of curves. Berger’s construction gives a rich class of

curves for which DPC does hold in every layer of a tower of coverings. We restate
Theorem 12.5 from Lecture 1 in a slightly different (but visibly equivalent) form.
Theorem 11.1. Let E be an elliptic curve over K = k(t) and assume that there
are rational functions f (x) and g(y) on P1k such that E is birational to the curve
V (f (x) − tg(y)) ⊂ P1K × P1K . Then the BSD conjecture holds for E over the field
k(u) = k(t1/d ) for all d prime to p.
Proof. Clearing denominators we may interpret f (x) − tg(y) as defining a
hypersurface X in the affine space A3 with coordinates x, y, and t and it is clear
that the elliptic surface E → P1 associated to E is birationally isomorphic to X .
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 261

On the other hand, X is visibly birational to P1 × P1 since we may eliminate t.

Thus X and E are dominated by a product of curves. This checks the case d = 1.
For larger d, note that the elliptic surface Ed → P1 associated to E/k(u) is
birational to the hypersurface Xd in A3k defined by f (x) − ud g(y). Berger showed
by a fundamental group argument, generalizing [Sch96], that Xd is dominated by a
product of curves, more precisely, by a product of covers of P1 . (For her argument
0
to be correct, π1 should be replaced by the prime-to-p fundamental group π1p
throughout.) This was later made more explicit in [Ulm09a], where it was observed
that Xd is dominated by a product of two explicit covers of P1 .
More precisely, let Cd and Dd be the covers of P1k defined by z d = f (x) and
wd = g(y). Then there is a rational map from Cd × Dd to the hypersurface Xd ,
namely
(x, z, y, w) 7→ (x, y, u = z/w).
This is clearly dominant and so Xd and E are dominated by products of curves.
Applying Theorem 9.1 finishes the proof. 
Note that there is a great deal of flexibility in the choice of data for Berger’s
construction. As an example, take f (x) = x(x − a)/(x − 1) and g(y) = y(y − 1)
where a ∈ Fq is a parameter. Then if a 6= 1, the curve f (x) = tg(y) in P1 × P1 has
genus 1 and a rational point. A simple calculation shows that it is birational to the
Weierstrass cubic
y 2 + txy − ty = x3 − tax2 + t2 ax.
Theorem 11.1 implies that this curve satisfies the BSD conjecture over Fqn (t1/d )
for all n and all d prime to p. Varying q and a we get infinitely many curves for
which BSD holds at every layer of a tower.
We will give more examples and discuss further applications of the idea behind
Berger’s construction in Lectures 4 and 5.
 LECTURE 4

Unbounded ranks in towers

In order to prove results on analytic ranks in towers, we need a more sophisti-

cated approach to L-functions. In this lecture we explain Grothendieck’s approach
to L-functions over function fields and then use it and a new linear algebra lemma
to find elliptic curves with unbounded analytic and algebraic ranks in towers of
function fields.

1. Grothendieck’s analysis of L-functions

1.1. Galois representations
As usual, we let K = k(C) be the function field of a curve over a finite field k and
GK = Gal(K sep /K) its Galois group. As in Lecture 0, Section 2, we write Dv , Iv ,
and Frv for the decomposition group, inertia group, and (geometric) Frobenius at
a place v of K.
We fix a prime ` 6= p and consider a representation
(1.1.1) ρ : GK → GL(V ) ∼= GLn (Q` )
on a finite-dimensional Q` vector space. We make several standing assumptions
about ρ.
First, we always assume ρ is continuous and unramified away from a finite set
of places of K. By a compactness argument (see [KS99, 9.0.7]) , it is possible to
define ρ over a finite extension L of Q` , i.e., there is a representation
ρ0 : GK → GLn (L)
isomorphic to ρ over Q` . Nothing we say will depend on the field of definition of
ρ and we will generally not distinguish between ρ and isomorphic representations
defined over subfields of Q` .
We also always assume that ρ is pure of integral weight w, i.e., for all v where
w/2
ρ is unramified, the eigenvalues of ρ(Frv ) are Weil numbers of size qv .
Finally, we sometimes assume that ρ is “symplectically self-dual of weight w.”
This means that on the space V where ρ acts, there is an GK -equivariant, alter-
nating pairing with values in Q` (−w).

1.2. Conductors
The Artin conductor of ρ is a divisor on C (a formal P sum of places of K) and is a
measure of its ramification. We write Cond(ρ) = n = v nv [v]. To define the local
coefficients, fix a place v of K and let Gi ⊂ Iv be the higher ramification groups at
v (in the lower numbering). Then define
∞
X 1
nv = dim V /V Gi .
i=0
[G0 : Gi ]
263
264 LECTURE 4. UNBOUNDED RANKS IN TOWERS

Here V Gi denotes the subspace of V invariant under Gi . It is clear that nv = 0 if

and only if ρ is unramified at v. If ρ is tamely ramified at v (i.e., G1 acts trivially),
then nv = dim V /V G0 = dim V /V Iv . In general, the first term of the sum above
is the tame conductor and the rest of the sum is the Swan conductor . We refer to
[Mil80, V.2] and also [Ser77, §19] for an alternative definition and more discussion
about the conductor, including the fact that the local coefficients nv are integers.

1.3. L-functions
Let us fix an isomorphism Q` ∼ = C so that we may regard eigenvalues of Frobenius
on `-adic representations as complex numbers. Having done this, a representa-
tion (1.1.1) gives rise to an L-function, defined as an Euler product:
Y
det 1 − T Frv |V Iv


(1.3.1) L(ρ, T ) =
v
−s
and L(ρ, s) = L(ρ, q ). The product is over the places of K, the exponent Iv
denotes the subspace of elements invariant under the inertia group Iv , and Frv is a
Frobenius element at v.
Because of our assumption that ρ is pure of weight w, the product defining
L(ρ, s) converges absolutely and defines a holomorphic function in the region Re s >
w/2 + 1.
It is clear from the definition that if ρ and σ are Galois representations then
L(ρ ⊕ σ, s) = L(ρ, s)L(σ, s) and L(ρ(n), s) = L(ρ, s − n).
It is also clear that L(ρtriv , s) = ζ(C, s). and so L(ρtriv (n), s) = ζ(C, s − n).
Exercise 1.3.2. Prove that if ρ factors through GK → Gk , so that Frv goes to
αdeg v , then
L(ρ, T ) = Z(C, αT )
is a twisted version of the zeta function of C. Compare with Exercise 9.2 of Lecture 1.
Note that a representation factors through GK → Gk if and only if it is trivial on
GkK , so this exercise fills in the missing cases in the following theorem.
Theorem 1.3.3. Suppose that ρ is a representation of GK (satisfying the standing
hypotheses of Subsection 1.1) that contains no copies of the trivial representation
when restricted to GkK . Then there is a canonically defined Q` -vector space H(ρ)
with continuous Gk action such that
L(ρ, s) = det 1 − q −s Frq |H(ρ) .

The dimension of H(ρ) is deg(ρ)(2gC − 2) + deg n where n is the conductor of ρ.

Proof. (Sketch) The representation ρ : GK → GL(V ) gives rise to a con-
structible sheaf Fρ on C. In outline: ρ is essentially the same thing as a lisse sheaf
FU on the open subset j : U ,→ C over which ρ is unramified. We defined Fρ as
the push-forward j∗ FU . For each closed point v of C, the stalk of ρ at v is V Iv .
Let H i (C, F) be the étale cohomology groups of F. They are finite dimensional
Q` vector spaces and give continuous representations of Gk .
The Grothendieck-Lefschetz fixed point formula says that for each finite exten-
sion Fqn of k ∼= Fq , we have
X 2
X
(−1)i Tr F rqn |H i (C, F) .


Tr(F rx |Fx ) =
x∈C(Fqn ) i=0
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 265

On the left hand side, the sum is over points of C with values in Fqn and the
summand is the trace of the action of the Frobenius at x on the stalk of F at a
geometric point over x.
Multiplying both sides by T n /n, summing over n ≥ 1, and exponentiating, one
finds that
2
Y
(−1)i+1
L(ρ, T ) = det 1 − T Frq |H i (C, F) .
i=0
Now H 0 (C, F) and H 2 (C, F) are isomorphic respectively to the invariants and
coinvariants of V under GkK and so under our hypotheses on ρ, H i (C, F) vanishes
for i = 0, 2. Thus we have
L(ρ, s) = det 1 − q −s Frq |H(ρ)

where H(ρ) = H 1 (C, F).

The dimension formula comes from an Euler characteristic formula proven by
Raynaud and sometimes called the Grothendieck-Ogg-Shafarevich formula. It says
2
X
(−1)i dim H i (C, F) = deg(ρ)(2 − 2gC ) − deg(Cond(ρ)).
i=0

Since H 0 and H 2 vanish, this gives the desired dimension formula. 

Obviously we have omitted many details. I recommend [Mil80, V.1 and V.2]
as a compact and readable source for several of the key points, including passing
from `-torsion sheaves to `-adic sheaves, the conductor, and the Grothendieck-
Ogg-Shafarevich formula. See [Mil80, VI.13] for the Grothendieck-Lefschetz trace
formula.
Remark/Exercise 1.3.4. If we are willing to use a virtual representation of Gk in
place of a usual representation, then the Theorem has a more elegant restatement
which avoids singling out representations that are trivial when restricted to GkK .
State and prove this generalization.
Exercise 1.3.5. Check that we have the Artin formalism formula: if F/K is a
finite separable extension and ρ is a representation of GF , then
L(ρ, s) = L(IndG
GF ρ, s).
K

Note that the left hand side is an Euler product on F with almost all factors of
some degree, say N , whereas the right hand side is an Euler product on K, with
almost all factors of degree N [F : K]. The equality can be taken to be an equality
of Euler products, where that on the left is grouped according to the places of K.
1.4. Functional equation and Riemann hypothesis
Theorem 1.3.3 shows that the L-function of ρ has an analytic continuation to the
entire s plane (meromorphic if we allow ρ to have trivial factors over kK). In this
section we deduce other good analytic properties of L(ρ, s).
Theorem 1.4.1. Suppose (in addition to the standing hypotheses) that ρ is sym-
plectically self-dual of weight w. Then L(ρ, s) satisfies a functional equation
L(ρ, w + 1 − s) = ±q N (s−(w+1)/2) L(ρ, s)
where N = (2gC − 2) deg(ρ) + deg(Cond(ρ)). The zeroes of ρ lie on the line Re s =
(w + 1)/2.
266 LECTURE 4. UNBOUNDED RANKS IN TOWERS

Proof. (Sketch) We use the notation of the proof of Theorem 1.3.3. The
functional equation comes from a symmetric pairing
∼ Q` (−w − 1).
H(ρ) × H(ρ) → H 2 (C, Q` (−w)) =
(Symmetric because ρ is skew-symmetric and H = H 1 .) That there is such a
pairing is not as straightforward as it looks, because we defined the sheaf F as
a push forward j∗ FU where j : U ,→ C is a non-empty open set over which ρ is
unramified and FU is the lisse sheaf on U corresponding to ρ. It is well-known that
j ∗ identifies H 1 (C, F) with the image of the “forget supports” map
Hc1 (U , FU ) → H 1 (U , FU )
from compactly supported cohomology to usual cohomology. (This is often stated,
but the only proof I know of in the literature is [Ulm05, 7.1.6].) The cup product
Hc1 (U , FU ) × H 1 (U , FU∗ ) → Hc2 (U , Q` ) ∼
= Q` (−1)
then induces a pairing on H 1 (C, F) via the above identification. Poincaré duality
shows that the pairing is non-degenerate and so H(ρ) is orthogonally self-dual of
weight w + 1.
The location of the zeroes is related to the eigenvalues of Frobenius on H(ρ) =
1
H (C, F) and these are Weil numbers of size q w+1 by Deligne’s purity theorem
[Del80]. I recommend the Arizona Winter School 2000 lectures of Katz (published
as [Kat01]) for a streamlined proof of Deligne’s theorem in the generality needed
here. 

2. The case of an elliptic curve

Next, we apply the results of the previous section to elliptic curves. Throughout,
E will be an elliptic curve over a function field K = k(C) over a finite field k of
characteristic p.

2.1. The Tate module

We consider the Tate module of E. More precisely, fix a prime ` 6= p and let
T` E = lim E(K)[`n ] and V` E = T` E ⊗Z` Q` .
← −
n

Let ρE be the representation of GK on the dual vector space V`∗ = Hom(V` E, Q` ) ∼ =

H 1 (E, Q` ). Then ρE is two-dimensional and continuous and (by the criterion of
Ogg-Néron-Shafarevich, see [ST68, Thm. 1]) it is unramified outside the (finite) set
of places where E has bad reduction.
At every place v of K where E has good reduction, we have
det(1 − ρ(Frv )T ) = 1 − av T + qv T 2
where av is defined as in (8.3) by #Ev (κv ) = 1 − av + qv . This follows from the
smooth base change theorem [Mil80, VI.4] and the cohomological description of the
zeta function of the reduction, as in Section 4 of Lecture 0. Thus ρ is pure of weight
w = 1.
The Weil pairing induces an alternating, Gk -equivariant pairing V` E × V` E →
Q` (−1) and so ρ is symplectically self-dual of weight 1.
If E is constant, then ρE factors through GK → Gk and since Gk is abelian,
ρE is the direct sum of two characters. More precisely, if E ∼ = E0 ×k K and
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 267

1 − aT + qT 2 = (1 − α1 T )(1 − α2 T ) is the numerator of the Z-function of E0 , then

ρE is the sum of the two characters that send Frv to αideg v .
If E is non-isotrivial, then ρE restricted to GkK has no trivial subrepresenta-
tions. One way to see this is to use a slight generalization of the MWLN theorem,
according to which E(kK) is finitely generated (when E is non-isotrivial). Thus its
`-power torsion is finite and this certainly precludes a trivial subrepresentation in
ρ|GkK . In fact, by a theorem of Igusa [Igu59], ρ|GkK is contains an open subgroup
of SL2 (Z` ) so is certainly irreducible, even absolutely irreducible.
Exercise 2.1.1. Show that if E is isotrivial but not constant, then ρE restricted
to GkK has no trivial subrepresentation. Hint: E is a twist of a contant curve
E 0 = E0 ×k K. Relate the action of GK on the Tate module of E to its action on
that of E 0 and show that there exists an element σ ∈ GkK that acts on V` E via a
non-trivial automorphism of E. But a non-trivial automorphism has only finitely
many fixed points.
We can summarize this discussion as follows.
Proposition 2.1.2. Let ρ be the action of GK on the Tate module V` E of E.
Then ρ is continuous, unramified outside a finite set of places of K, and is pure
and symplectically self-dual of weight 1. If E is non-constant, then ρ|GkK has no
trivial subrepresentations.
The conductor of ρE as defined in the previous section is equal to the conductor
of E as mentioned in Section 8 of Lecture 1. This was proven by Ogg in [Ogg67].
2.2. The L-function
Applying the results of the previous section, we get a very satisfactory analysis
of the L-function of E. Since we know everything about the constant case by an
elementary analysis (cf. exercise 9.2 of Lecture 1), we restrict to the non-constant
case.
Theorem 2.2.1. Let E be a non-constant elliptic curve over K = k(C) and let q
be the cardinality of k. Let n be the conductor of E. Then L(E, s) is a polynomial
in q −s of degree N = 4gC − 4 + deg(n). Its inverse roots are Weil numbers of size
q and it satisfies a functional equation
L(E, 2 − s) = ±q N (s−1) L(E, s).
Combining the Theorem with Theorem 12.1, we obtain the following.
Corollary 2.2.2. The rank of E(K) is bounded above by N = 4gC − 4 + deg(n). If
equality holds, then L(E, s) = (1 − q 1−s )N .
The sign in the functional equation can be computed as a product of local
factors. This can be seen via the connection with automorphic forms (a connection
which is outside the scope of these lectures) or, because we are in the function field
situation, directly via cohomological techniques. See [Lau84] for the latter.

3. Large analytic ranks in towers

3.1. Statement of the theorem
We give a general context in which one obtains large analytic ranks by passing to
layers of a suitable tower of function fields.
268 LECTURE 4. UNBOUNDED RANKS IN TOWERS

As usual, let p be a prime and q a power of p. Let K = Fq (t), for each d not
divisible by p, set Fd = Fq (t1/d ) ∼
= Fq (u), and Kd = Fq (µd )(t1/d ) ∼
= Fq (µd )(u).
Suppose that E is an elliptic curve over K. Let n be the conductor of E and
let
n0 = n − dim(V` E/V` E I0 )[0] − dim(V` E/V` E I∞ )[∞].
This is the conductor of E except that we have removed the tame part at t = 0
and t = ∞.
Theorem 3.1.1. Let E be an elliptic curve over K and define n0 as above. Suppose
that deg n0 is odd. Then the analytic rank of E over Fd (and Kd ) is unbounded as
d varies. More precisely, there exists a constant c depending only on E such that if
d has the form d = q n + 1, then
d qn + 1
ords=1 L(E/Fd , s) ≥ −c= − c.
2n 2n
and
ords=1 L(E/Kd , s) ≥ d − c = q n + 1 − c
This theorem is proven in detail in [Ulm07, §2-4]. We will sketch the main lines
of the argument below.

3.2. A linear algebra lemma

Our analytic rank results ultimately come from the following odd-looking result of
linear algebra.
Proposition 3.2.1. Let V be a finite-dimensional vector space with subspaces Wi
indexed by i ∈ Z/aZ such that V = ⊕i∈Z/aZ Wi . Let φ : V → V be an invertible
linear transformation such that φ(Wi ) = Wi+1 for all i ∈ Z/aZ. Suppose that V
admits a non-degenerate, φ-invariant symmetric bilinear form h, i. Suppose that
a is even and h, i induces an isomorphism Wa/2 ∼= W0∗ (the dual vector space of
W0 ). Suppose also that N = dim W0 is odd. Then the polynomial 1 − T a divides
det(1 − φT |V ).
We omit the proof of this proposition, since it is not hard and it appears in
two forms in the literature already. Namely, embedded in [Ulm05, 7.1.11ff] is a
matrix-language proof of the proposition, and a coordinate-free proof is given in
[Ulm07, §2].

3.3. Sketch of the proof of Theorem 3.1.1

For simplicity, we assume that E is non-isotrivial. (If p > 3 and E is isotrivial, then
the theorem is vacuous because all of the local conductor exponents nv are even.)
Let ρ be the representation of GK on V = H 1 (E, Q` ) = (V` E)∗ and let ρd be the
restriction of ρ to GFd . Then by Grothendieck’s analysis, we have
L(E/Fd , s) = det 1 − Frq q −s |H(ρd ) .

Here H(ρd ) is an H 1 on the rational curve whose function field is Fd = Fq (u) =

Fq (t1/d ).
The projection formula in cohomology (a parallel of the Artin formalism 1.3.5)
implies that
H(ρd ) ∼
= H(IndG ∼ GK
GF ρ) = H(ρ ⊗ IndGF 1)
K
d d
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 269

where 1 denotes the trivial representation. Since the cohomology H is computed on

1 1 1
Pu (the P1 with coordinate u, with scalars extended to Fq ) and Pu → Pt is Galois
with group µd , we have
d−1
H(ρd ) ∼
M
= H(ρ ⊗ χj )
j=0

where χ is a character of Gal(Fq (u)/Fq (t)) of order exactly d.

Now the decomposition displayed above is not preserved by Frobenius. Indeed
Frq sends H(ρ ⊗ χj ) to H(ρ ⊗ χqj ). Thus we let o ⊂ Z/dZ denote an orbit for
multiplication by q and we regroup:
 

H(ρd ) ∼
M M
=  H(ρ ⊗ χj ) .
o⊂Z/dZ j∈o

We write Vo for the summand indexed by an orbit o ⊂ Z/dZ in the last display
and ao for the cardinality of o. As we will see presently, the hypotheses of the
theorem imply that Proposition 3.2.1 applies to most of the Vo and for each one
where it does, we get a zero of the L-function. Before we do that, there is one small
technical point to take care of: The linear algebra proposition requires that V be
literally self-dual (not self-dual with a weight) and it implies that 1 is an eigenvalue
of φ on V . To get the eigenvalue q that we need, we should twist ρ by −1/2 (which
is legitimate once we have fixed choice of square root of q) so that it has weight 0,
apply the lemma, and twist back to get the desired zero. We leave the details of
these points to the reader.
Assuming we have made the twist just mentioned, we need to check which Vo
are self-dual. Since ρ is self-dual, Poincaré duality gives a non-degenerate pairing on
H(ρd ) which puts H(ρ⊗χj ) in duality with H(ρ⊗χ−j ). Thus if d = q n +1 for some
n > 0, then all of the orbits o will yield a self-dual Vo . Possibly two of these orbits
have odd order (those through 0 and d/2, which have order 1) and all of the other
i
have ao even. Moreover, for the orbits of even order, setting Wo,i = H(ρ ⊗ χq jo )
for some fixed jo ∈ o, we have
o −1
aM
Vo ∼
= Wo,i
i=0
with Wo,i and Wo,i+a0 /2 in duality.
The last point that we need is that Wo,i should be odd-dimensional. The hy-
pothesis on n0 implies that for all characters χj of sufficiently high order (depending
only on E), the conductor of ρ ⊗ χj is odd. The Grothendieck-Ogg-Shafarevich di-
mension formula (mentioned at the end of the proof of Theorem 1.3.3) then implies
that for all orbits o consisting of characters of high order, H(ρ ⊗ χjo ) has odd
dimension.
The linear algebra proposition 3.2.1 now implies that for d = q n + 1 and for
most orbits o ⊂ Z/dZ, 1 is an eigenvalue of Frq on Vo (and q is an eigenvalue of Frq
on the corresponding factor of H(ρd )). Since each of these orbits has size ≤ 2n,
there is a constant c such that the number of “good” orbits is ≥ d/2n. Thus
d
ords=1 L(E/Fd , s) ≥ −c
2n
for a constant c depending only on E.
270 LECTURE 4. UNBOUNDED RANKS IN TOWERS

To get the assertions over Kd , note that in passing from Fd to Kd , each factor
(1 − q ao T ao ) of L(E/Fd , T ) becomes (1 − qT )ao and so
ords=1 L(E/Kd , s) ≥ d − c
for another c independent of E.
This completes our discussion of Theorem 3.1.1. We refer to [Ulm07, §2-4] for
more details. 

3.4. Examples
It is easy to see that the hypotheses in Theorem 3.1.1 are not very restrictive and
that high analytic ranks are in a sense ubiquitous. The following rephrasing of the
condition in the theorem should make this clear.
Exercise 3.4.1. Prove that if p > 3 and E is an elliptic curve over K, then
Theorem 3.1.1 guarantees that E has unbounded analytic rank in the tower Fd if
the number of geometric points of P1Fq over which E has multiplicative reduction is
odd.
Corollary 3.4.2. Let p be any prime number, K = Fp (t), and let E be one of the
curves E7 , E8 , or E9 defined in Subsection 1.2 of Lecture 1. Then
ords=1 L(E/Fp (t1/d ), s)
is unbounded as d varies through integers prime to p
Proof. If p > 3, then one sees immediately by considering the discriminant
and j-invariant that E has one finite, non-zero place of multiplicative reduction and
is tame at 0 and ∞, thus it satisfies the hypotheses of Theorem 3.1.1. If p = 2 or 3,
one checks using Tate’s algorithm that E has good reduction at all finite non-zero
places and is tame at zero, but the wild part of the conductor at ∞ is odd and so
the theorem again applies. 

For another example, take the Legendre curve

y 2 = x(x − 1)(x − t)
over Fp (t), p > 2. It is tame at 0 and ∞ and has exactly one finite, non-zero place
of multiplicative reduction.

4. Large algebraic ranks

4.1. Examples via the four-monomial theorem
Noting that the curves E7 , E8 , and E9 are defined by equations involving exactly
four monomials, we get a very nice result on algebraic ranks.
Theorem 4.1.1. Let p be any prime number, K = Fp (t), and let E be one of the
curves E7 , E8 , or E9 defined in Subsection 1.2 of Lecture 1. Then for all d prime
to p and all powers q of p, the Birch and Swinnerton-Dyer conjecture holds for E
over Kd = Fq (t1/d ). Moreover, the rank of E(Fp (t1/d )) is unbounded as d varies.
Proof. This follows immediately from Corollary 3.4.2 and Theorem 12.4 of
Lecture 1 as soon as we note that E/Kd is defined by an equation satisfying Shioda’s
conditions. 
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 271

Similar ideas can be used to show that for every prime p and every genus g > 0,
there is an explicit hyperelliptic curve C over Fp (t) such that the Jacobian of C
satisfies BSD over Fq (t1/d ) for all q and d and has unbounded rank in the tower
Fp (t1/d ). This is the main theorem of [Ulm07].
4.2. Examples via Berger’s construction
As we pointed out in Lecture 3, the Shioda 4-monomial construction is rigid—
varying the coefficients does not lead to families that vary geometrically. Berger’s
thesis developed a new construction with parameters that leads to families of curves
for which the BSD conjecture holds in a tower of fields. This together with the ana-
lytic ranks result 3.1.1 gives examples of families of elliptic curves with unbounded
ranks.
To make this concrete, we quote the first example with parameters from [Ber08]
that, together with the analytic rank construction 3.1.1, gives rise to unbounded
analytic and algebraic ranks.
Theorem 4.2.1 (Berger). Let k = Fq be a finite field of characteristic p and let
a ∈ Fq with a 6= 0, 1, 2. Let E be the elliptic curve over K = Fq (t) defined by
y 2 + a(t − 1)xy + a(t2 − t)y = x3 + (2a + 1)tx2 + a(a + 2)t2 x + a2 t3 .
Then for all d prime to p the BSD conjecture holds for E over Fq (t1/d ). Moreover,
for every q and a as above, the rank of E(Fq (t1/d )) is unbounded as d varies.
Proof. This is an instance of Berger’s construction (Theorem 11.1 of Lec-
ture 3). Indeed, let f (x) = x(x − a)/(x − 1) and g(y) = y(y − a)/(y − 1). Then
V (f − tg) ⊂ P1K × P1K is birational to E, which is a smooth elliptic curve for all
a 6= 0, 1. Berger’s Theorem 11.1 of Lecture 3 shows that E satisfies BSD over the
fields Fq (t1/d ).
The discriminant of E is
∆ = a2 (a − 1)4 t4 (t − 1)2 a2 t2 − (2a2 − 16a + 16)t + a2 .

Assume first that p > 3. One checks that ∆ is relatively prime to c4 so that the
zeroes of ∆ are places of multiplicative reduction. Since the discriminant (in t) of
the quadratic factor a2 t2 − (2a2 − 16a + 16)t + a2 is −64(a − 1)(a − 2)2 we see that
there are three finite, non-zero geometric points of multiplicative reduction. Since
p > 3, the reduction at 0 and ∞ is tame and so n0 (defined as in Subsection 3.1 of
Lecture 4) has degree 3. Thus by Theorem 3.1.1 of Lecture 4, E has unbounded
analytic ranks in the tower Fq (t1/d ) and thus also unbounded algebraic ranks by
the previous paragraph on BSD.
If p = 2 or 3, one needs to use Tate’s algorithm to compute n0 , which again
turns out to have degree 3. We leave the details of this computation as a pleasant
exercise for the reader. 
 LECTURE 5

More applications of products of curves

In the last part of Lecture 4, we chose special curves E and used a domination
C ×D99KE of the associated surface to deduce the Tate conjecture for E and thus the
BSD conjecture for E. This yields an a priori equality of analytic and algebraic
ranks. We then used other, cohomological, methods (namely the analytic ranks
theorem) to compute the analytic rank.
It turns out to be possible to use domination by a product of curves and geom-
etry to prove directly results about algebraic ranks and explicit points. We sketch
some of these applications in this lecture.

1. More on Berger’s construction

Let k be a field (not necessarily finite), K = k(t), and Kd = k(t1/d ) = k(u). Recall
that in Berger’s construction we start with rational curves C = P1k and D = P1k
and rational functions f (x) on C and g(y) on D. We get a curve in P1K × P1K
defined by f (x) − tg(y) = 0 and we let E be the smooth proper model over K of
this curve. (Some hypotheses are required for this to exist, but they are weaker
than our standing hypotheses below.) The genus of E was computed by Berger in
[Ber08, Theorem 3.1]. All the examples we consider will be of genus 1 and will have
a K-rational point.
We establish more notation to state a precise result. Let us assume for sim-
plicity all the zeroes and poles of f and g are k-rational. Write
0 0
k
X k
X `
X `
X
(1.1) div(f ) = a i Pi − a0i0 Pi00 and div(g) = bj Qj − b0j 0 Q0j 0
i=1 i0 =1 j=1 j 0 =1

with ai , a0i0 , bi , b0j 0 positive integers and Pi , Pi00 , Qj , and Q0j 0 distinct k-rational
points. Let
0 0
k
X k
X `
X `
X
m= ai = a0i0 and n = bj = b0j 0 .
i=1 i0 =1 j=1 j 0 =1

As standing hypotheses, we assume that: (i) all the multiplicities ai , a0i0 , bj ,

and b0j 0 are prime to the characteristic of k; and (ii) gcd(a1 . . . , ak , a01 , . . . , a0k0 ) =
gcd(b1 . . . , b` , b01 , . . . , b0`0 ) = 1.
Under these hypotheses, Berger computes that the genus of E is
X X
(1.2) gE = (m − 1)(n − 1) − δ(ai , bj ) − δ(a0i0 , b0j 0 )
i,j i0 ,j 0

where δ(a, b) = (ab − a − b + gcd(a, b))/2.

From now on we assume that we have chosen the data f and g so that E has
genus 1. Two typical cases are where f and g are quadratic rational functions with
273
274 LECTURE 5. MORE APPLICATIONS OF PRODUCTS OF CURVES

simple zeroes and poles, or where f and g are cubic polynomials. There is always a
K-rational point on E; for example, we may take a point where x and y are zeroes
of f and g.
Let Ed → P1 be the elliptic surface over k attached to E/Kd . It is clear that
Ed is birational to the closed subset of P1k × P1k × P1k (with coordinates x, y, u)
defined by the vanishing of f (x) − ud g(y). We saw in Section 11 of Lecture 3 that
E is dominated by a product of curves and we would now like to make this more
precise.
Recall that we defined covers Cd → C = P1 and Dd → D = P1 by the equations
z = f (x) and wd = g(y). Note that there is an action of µd , the d-th roots of
d

unity, on Cd and on Dd .
Proposition 1.3. The surface Ed is birationally isomorphic to the quotient surface
(Cd × Dd )/µd where µd acts diagonally.
Proof. We have already noted that Ed is birational to the zero set X of f (x)−
ud g(y) in P1k × P1k × P1k . Define a rational map from Cd × Dd to X by sending
(x, z, y, w) to (x, y, u = z/w). It is clear that this map factors through the quotient
(Cd × Dd )/µd . Since the map is generically of degree d, it induces a birational
isomorphism between (Cd × Dd )/µd and X . Thus (Cd × Dd )/µd is birationally
isomorphic to Ed . 
In the next section we will explain how this birational isomorphism can be used
to compute the Néron-Severi group of Ed and the Mordell-Weil group E(Kd ).

2. A rank formula
We keep the notation and hypotheses of the preceding subsection. Consider the
base P1k , the one corresponding to K, with coordinate t. For each geometric point
x of this P1k , let fx be the number of components in the fiber of E → P1 over x.
For almost all x, fx = 1 and its value at any point can be computed using Tate’s
algorithm.
Define two constants c1 and c2 by the formulae
X
c1 = (fx − 1)
x6=0,∞

and
c2 = (k − 1)(` − 1) + (k 0 − 1)(`0 − 1).
Here the sum is over geometric points of P1k except t = 0 and t = ∞ and k, k 0 , `,
and `0 are the numbers of distinct zeroes and poles of f and g (cf. equation (1.1)).
Note that c1 and c2 depend only on the data defining E/K, not on d.
Theorem 2.1. Suppose that k is algebraically closed and that d is relatively prime
to all of the multiplicities ai , a0i0 , bj , and b0j 0 and to the characteristic of k. Then
we have
Rank E(Kd ) = Rank Hom(JCd , JDd )µd − c1 d + c2 .
µd
Here Hom(· · · ) signifies the homomorphisms commuting with the actions of µd
on the two Jacobians induced by its action on the curves.
Sketch of Proof. In brief, we use the birational isomorphism
(Cd × Dd )/µd 99KEd
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 275

to compute the rank of the Néron-Severi group of Ed and then use the Shioda-Tate
formula to compute the rank of E(Kd ).
More precisely, we saw in Lecture 2, Subsection 8.4 that the Néron-Severi group
of the product Cd × Dd is isomorphic to Z2 × Hom(JCd , JDd ). It follows easily
that the Néron-Severi group of the quotient (Cd × Dd )/µd is isomorphic to Z2 ×
Hom(JCd , JDd )µd .
One then keeps careful track of the blow-ups needed to pass from (Cd × Dd )/µd
to Ed . The effect of blow-ups on Néron-Severi is quite simple and was noted in
Subsection 8.5 of Lecture 2. This is the main source of the term c2 in the formula.
Finally, one computes the rank of E(Kd ) using the Shioda-Tate formula, as in
Section 5 of Lecture 3. This step is the main source of the term c1 d.
The hypothesis that k is algebraically closed is not essential for any of the above,
but it avoids rationality questions that would greatly complicate the formula.
For full details on the proof of this theorem (in a more general context) see
[Ulm09a, Section 6]. 

3. First examples
One of the first examples is already quite interesting. We give a brief sketch and
refer to [Ulm09a] for more details.
With notation as in Section 1, we take f (x) = x(x − 1) and g(y) = y 2 /(1 − y).
The genus formula (1.2) shows that E has genus 1. In fact, the change of coordinates
x = −y/(x + t), y = −x/t brings it into the Weierstrass form
y 2 + xy + ty = x3 + tx2 .
We remark in passing that if the characteristic of k is not 2, E has multiplicative
reduction at t = 1/16 and good reduction elsewhere away from 0 and ∞. Thus by
the analytic rank result of Lecture 2, when k is finite, say k = Fp and p > 3, we
expect E to have unbounded analytic rank in the tower Fp (t1/d ). (In fact a more
careful analysis gives the same conclusion for every p.)
Now assume that k is algebraically closed. To compute the constant c1 , one
checks that (for k of any characteristic) E has exactly one irreducible component
over each geometric point of P1k . Thus c1 = 0. It is immediate from the definition
that c2 = 0. Thus our rank formula yields
Rank E(Kd ) = Rank Hom(JCd , JDd )µd .
Next we note that there is an isomorphism φ : Cd → Dd sending (x, z) to
(y = 1/x, w = 1/z). This isomorphism anti-commutes with the µd action: Let ζd
be a primitive d-th root of unity and write [ζd ] for its action on curves or Jacobians.
Then φ ◦ [ζd ] = [ζd−1 ] ◦ φ. Using φ to identify Cd and Dd , our rank formula becomes

Rank E(Kd ) = Rank End(JCd )anti−µd

where “End(· · · )anti−µd ” denotes those endomorphisms anti-commuting with µd in
the sense above.
Suppose that k has characteristic zero. Then a consideration of the (faithful)
action of End(JCd ) on the differentials H 0 (JCd , Ω1 ) shows that End(JCd )anti−µd = 0
for all d (see [Ulm09a, 7.6]). We conclude that for k of characteristic zero, the rank
of E(Kd ) is zero for all d.
276 LECTURE 5. MORE APPLICATIONS OF PRODUCTS OF CURVES

Now assume that k has characteristic p (and is algebraically closed). If we take

d of the form pf + 1 then we get many elements of End(JCd )anti−µd . Namely, we
consider the Frobenius Frpf and compute that
f
Frpf ◦[ζd ] = [ζdp ] ◦ Frpf = [ζd−1 ] ◦ Frpf .
The same computation shows that Frpf ◦[ζdi ] anticommutes with µd for all i. It
turns out that there are two relations among these endomorphism in End(JCd ) if
p > 2 and just one relation if p = 2 (see [Ulm09a, 7.8-7.10]). Thus we find that, for
d of the special form d = pf + 1,
(
d − 2 if p > 2
Rank E(Fp (t1/d )) =
d − 1 if p = 2.
The reader may enjoy checking that this is in exact agreement with what the
analytic rank result (Theorem 3.1.1 of Lecture 4) predicts.
Somewhat surprisingly, there are more values of d for which we get high ranks.
A natural question is to identify all pairs (p, d) such that E(Fp (t1/d ) has “new”
rank, i.e, points of infinite order not coming from smaller values of d. The exact
set of pairs (p, d) for which we get high rank is mysterious. There are “systematic”
cases (such as (p, pf + 1), as above, or (p, 2(p − 1))) and other cases that may be
sporadic. This is the subject of ongoing research so we will not go into more detail,
except to note that the example in Section 5 below is relevant to this question.

4. Explicit points
The main ingredients in the rank formula of Section 2 are the calculation of the
Néron-Severi group of a product of curves in terms of homomorphisms of Jacobians
and the Shioda-Tate formula. Tracing through the proof leads to a homomorphism
L1 NS(Ed ) ∼
Hom(JCd , JDd )µd ∼
= DivCorr(Cd , Dd ) → L1 NS(Ed ) → 2 = E(Kd ).
L NS(Ed )
For elements of Hom(JCd , JDd )µd where we can find an explicit representation
in DivCorr(Cd , Dd ), the geometry of Berger’s construction leads to explicit points
in E(Kd ). This applies notably to the endomorphisms Frpf ◦[ζdi ] appearing in the
analysis of the first example above. Indeed, these endomorphisms are represented
in DivCorr(Cd , Dd ) by the graphs of Frobenius composed with the automorphisms
[ζdi ] of Cd .
Tracing through the geometry leads to remarkable explicit expressions for
points in E(Kd ). The details of the calculation are presented in [Ulm09a, §8]
so we will just state the results here, and only in the case p > 2.
Theorem 4.1. Let p > 2, k = Fp and K = k(t). Let E be the elliptic curve
y 2 + xy + ty = x3 + tx2
over K. Let q = pf , d = q + 1, Kd = k(t1/d ), and
 q q
u (u − u) u2q (1 + 2u + 2uq ) u2q

P (u) = , − .
(1 + 4u)q 2(1 + 4u)(3q−1)/2 2(1 + 4u)q−1
Then the points Pi = P (ζdi t1/d ) for i = 0, . . . , d − 1 lie in E(Kd ) and they generate
a finite index subgroup of E(Kd ), which has rank d − 2. The relations among them
Pd−1 Pd−1
are that i=0 Pi and i=0 (−1)i Pi are torsion.
 DOUGLAS ULMER, ELLIPTIC CURVES OVER FUNCTION FIELDS 277

It is elementary to check that the points lie in E(Kd ). To check their indepen-
dence and the relations by elementary means, one may compute the height pairing
on the lattice they generate. It turns out to be a scaling of the direct sum of
two copies of the A∗(d−2)/2 lattice. Since we know from the previous section that
E(Kd ) has rank d − 2, the explicit points generate a subgroup of finite index. As
another check that they have finite index, we could compute the conductor of E—it
turns out to have degree d + 2—and apply Corollary 2.2.2 of Lecture 4. All this is
explained in detail in [Ulm09a, §8].

5. Another example
We keep the notation and hypotheses of Sections 1 and 2. For another example,
assume that k = Fp with p > 2. Let f (x) = x/(x2 − 1) and g(y) = y(y − 1). The
curve f (x)−tg(y) = 0 has genus 1 and the change of coordinates x = (x0 +t)/(x0 −t),
y = −y 0 /2tx0 brings it into the Weierstrass form
y 02 + 2tx0 y 0 = x03 − t2 x0 .
This curve, call it E, has multiplicative reduction of type I1 at the places dividing
t2 + 4, good reduction at other finite, non-zero places, and tame reduction at t = 0
and t = ∞. We find that the constants c1 and c2 are both zero and that
Rank E(Fp (t1/d )) = Rank Hom(JCd , JDd )µd .
Recall that the curves Cd and Dd are defined by the equations
x
z d = f (x) = 2 and wd = g(y) = y(y − 1).
x −1
Consider the morphism φ : Cd → Dd defined by φ∗ (y) = 1/(1 − x2 ) and φ∗ (w) = z 2 .
It is obviously not constant and so induces a surjective homomorphism φ∗ : JCd →
JDd .
The homomorphism φ∗ clearly does not commute with the action of µd . Indeed,
if ζd denotes a primitive d-th root of unity and [ζd ] its action on one of the Jacobians,
we have φ∗ ◦ [ζd ] = [ζd2 ] ◦ φ∗ . (This formula already holds at the level of the curves
Cd and Dd .)
Now let us assume that d has the form d = 2pf − 1 and consider the map
φ ◦ Frpf : Cd → Dd . Then we find that
f
(φ ◦ Frpf )∗ ◦ [ζd ] = [ζd2p ] ◦ (φ ◦ Frpf )∗ = [ζd ] ◦ (φ ◦ Frpf )∗
in Hom(JCd , JDd ), in other words that (φ ◦ Frpf )∗ commutes with the µd action.
Similarly ([ζdi ] ◦ φ ◦ Frpf )∗ commutes with the µd action for all i.
Further analysis of the homomorphisms ([ζdi ] ◦ φ ◦ Frpf )∗ in Hom(JCd , JDd )µd
(along the lines of [Ulm09a, 7.8]) shows that they are almost independent; more
precisely, they generate a subgroup of rank d − 1. Thus we find (for d of the form
d = 2pf − 1) that the rank of E(k(t1/d )) is at least d − 1.
The reader may find it a pleasant exercise to write down explicit points in this
situation, along the lines of the discussion in Section 4 and [Ulm09a, §8].

6. Further developments
There have been further developments in the area of rational points on curves and
Jacobians over function fields. To close, we mention three of them.
278 LECTURE 5. MORE APPLICATIONS OF PRODUCTS OF CURVES

In the examples of Sections 3 and 5, the set of d that are “interesting,” i.e., for
which we get high rank over Kd , depends very much on p, the characteristic of k.
In his thesis (University of Arizona, 2010), Tommy Occhipinti gives, for every p,
remarkable examples of elliptic curves E over Fp (t) such that for all d prime to p
we have
Rank E(Fp (t1/d )) ≥ d.
The curves come from Berger’s construction where f and g are generic degree two
rational functions. The rank inequality comes from the rank formula in Theorem 2.1
and the Honda-Tate theory of isogeny classes of abelian varieties over finite fields.
In the opposite direction, the author and Zarhin have given examples of curves
of every genus over C(t) such that their Jacobians have bounded rank in the tower
n
of fields C(t1/` ) where ` is a prime. See [UZ10].
Finally, after some encouragement by Dick Gross at PCMI, the author produced
explicit points on the Legendre curve over the fields Fp (µd )(t1/d ) where d has the
form pf + 1 and proved in a completely elementary way that they give Mordell-Weil
groups of unbounded rank. In fact, this construction is considerably easier than
that of Tate and Shafarevich [TS67] and could have been found in the 1960s. See
[Ulm09b].
It appears that this territory is rather fertile and that there is much still to
be discovered about high ranks and explicit points on curves and Jacobians over
function fields. Happy hunting!
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