Eisenstein cocycles for imaginary quadratic fields

Emmanuel Lecouturier, Romyar Sharifi, Sheng-Chi Shih, Jun Wang

Abstract
We construct Eisenstein cocycles for arithmetic subgroups of GL2 of imaginary
quadratic fields valued in second K-groups of products of two CM elliptic curves. We use
these to construct maps from the first homology groups of Bianchi spaces to corresponding
second K-groups of ray class fields and to verify the Eisenstein property of these maps
for prime-to-level Hecke operators.

1 Introduction
1.1 Background and the main theorem
The papers [Bus] and [Sha] defined explicit maps from first homology groups of modular
curves to second K-groups of cyclotomic integer rings. For a positive integer N , the map has
the form
Π◦N : H1 (X1 (N ), C, Z[ 12 ])+ → (K2 (Z[µN , N1 ]) ⊗ Z[ 21 ])+ ,
where C is the set of cusps not lying over ∞ ∈ X0 (N ), and a superscript “+” denotes fixed
part under complex conjugation (or a projection to said part). It carries explicit generators of
homology [u : v] for u, v ∈ Z/N Z − {0} with (u, v) = 1, known as Manin symbols [Ma], to
Steinberg symbols of cyclotomic N -units via the explicit recipe

Π◦N ([u : v]+ ) = {1 − ζNu , 1 − ζNv }+ ,

where ζN is a fixed primitive N th root of unity. A map from homology to a formally defined
and infinite but related group had been considered by Goncharov in [Gon1, Section 4].
The second author conjectured that Π◦N should be Eisenstein in the sense that

Π◦N ◦ (Tℓ − ℓ − ⟨ℓ⟩) = 0 (1.1)

for all primes ℓ ∤ N and

Π◦N ◦ (Uℓ∗ − 1) = 0

1
for all primes ℓ | N , where Uℓ∗ denotes the dual Hecke operator. Fukaya and Kato proved this
for the projection of Π◦N to the p-part of the second K-group for each p | N by realizing Π◦N
as the specialization at infinity of a Hecke-equivariant zeta map [FuKa, Theorem 5.3.3].
The restriction of Π◦N provides a map

ΠN : H1 (X1 (N ), Z[ 21 ])+ → (K2 (Z[µN ]) ⊗ Z[ 21 ])+ ,

which is then conjecturally also Eisenstein. Moreover, the second author has conjectured that
the map induced by ΠN on its quotient by a corresponding Eisenstein ideal is actually an
isomorphism (see [Sha, ShVe]). On the other hand, Π◦N is not an isomorphism for certain N .
In [ShVe], the second author and Venkatesh proved that ΠN is Eisenstein away from the
level N , i.e., satisfies (1.1) for ℓ ∤ N . To do so, they realized ΠN as a specialization at an
N -torsion point of the restriction to Γ1 (N ) of a cocycle

ΘN : GL2 (Z) → K2 (Q(G2m ))/⟨{−z1 , −z2 }⟩,

where zi is the ith coordinate function on G2m . The work of the first and fourth authors [LeWa]
combines the results of [FuKa] and [ShVe] together with level compatibilities to obtain the full
Eisenstein property of ΠN upon inverting 3.
In [FKS, Section 4.2], Fukaya, Kato, and the second author directly raised the question
that had been floating out there as to whether an analogous Eisenstein map exists between the
homology of a Bianchi space and the second K-group of a ray class field of an imaginary
quadratic field F , with elliptic units replacing cyclotomic units. Here is a representative
theorem from Section 5.1.

Theorem. Let F be an imaginary quadratic field with integer ring O, and let N be a nonzero
ideal of O such that the canonical map O× → (O/N )× is injective. Let Y1 (N ) be the Bianchi
space for F with Γ1 (N )-level structure. Let c be a proper ideal of O prime to N . Let O′ (N )
denote the ring of integers O(N ) of the ray class field of F of modulus N if N is not a prime
power and O(N )[ N1 ] otherwise. Then there exists a homomorphism

c ΠN
1
: H1 (Y1 (N ), Z[ 30 ]) → K2 (O′ (N )) ⊗ Z[ 30
1
]

that is Eisenstein away from N , which is a specialization at N -torsion of an Eisenstein tuple

of cocycles valued in second K-groups of products of two elliptic curves with CM by O. Given
a second proper ideal d prime to N , we have

(N d2 − R(d)) ◦ c ΠN = (N c2 − R(c)) ◦ d ΠN

for N the absolute norm and R the Artin reciprocity map on the ray class group of modulus
N.

2
 Of course, were it not for the imprecise caveat about its definition as a specialization, the
zero map would automatically satisfy the theorem, but we do not define c ΠN as such! Though
we have not attempted it, to see that c ΠN is nonzero should be possible in certain cases by
comparison with the maps ΠN for modular curves, which can often be shown to be nonzero.
In Theorem 5.2.9, we show under certain conditions that c ΠN factors through the homology
of the Satake compactification X1 (N ) given by adjoining the cusps and that c ΠN arises from
a map ΠN independent of c, but without connecting it to any expected explicit formulas.
The idea that a map ΠN should exist, sans its Eisenstein property, was hinted at by
Goncharov’s construction of a related map to a Q-vector space in [Gon2, Theorem 5.7] for
prime level and F ∈ {Q(i), Q(µ3 )}. The work of Calegari and Venkatesh [CaVe, Theorem
4.5.1(ii)] addressed the Eisenstein property: for F such that H1 (Y0 (1), Z) is torsion and prime
q of O, they constructed a surjective map from a subgroup of H1 (Y0 (q), Z) to K2 (O) ⊗ Z[ 61 ]
and showed it factors through the Eisenstein quotient. Computations of homology groups of
Bianchi spaces modulo Eisenstein ideals were performed in the Ph.D. thesis of Powell [Pow]
as a test case for the speculations of [FKS].
There have recently been a variety of constructions of what are commonly referred to as
Eisenstein cocycles. Among them are the Sczech-style cocycles of Bergeron, Charollois, and
Garcia [BCG] and Flórez, Karabulut, and Wong [FKW] for GLn over an imaginary quadratic
field and the equivariant coherent classes of Kings and Sprang [KiSp] over CM fields. These
fascinating works aim at formulas for critical values of L-functions of Hecke characters related
to Eisenstein-Kronecker series and Dedekind sums. There exist tentative connections between
these works and ours, but the settings, methods, and aims are markedly different.

1.2 Outline of the construction

To construct our maps, we refine the approach of [ShVe]. Let us outline this, highlighting a
variety of issues we must overcome. Just as cyclotomic units are the specializations of the
function 1 − z at z at roots of unity, elliptic units are specializations of theta functions on a
CM elliptic curve at a torsion point. With this in mind, and similarly to the case of a square of
an elliptic curve considered in [ShVe], we replace G2m by a product E × E ′ over L of two CM
elliptic curves over an abelian extension L of an imaginary quadratic field F .
Much as the case of G2m , our cocycle is constructed using a two-term Gersten-Kato complex
K = [K2 → K1 → K0 ] of the form
M M
K2 (L(E × E ′ )) → L(D)× → Z
D x

with the sums taken over irreducible divisors D and zero-cycles x on E × E ′ , respectively.

3
Specifically, we use the trace-fixed subcomplex K(0) of K, after inverting 5!. This fits in an
exact sequence
(0) (0) (0) 1
0 → K2 → K1 → K0 → Z[ 30 ] → 0,
the last map being the degree map on zero-cycles. Let c be a proper, nonzero ideal of the ring
of integers O of F . Analogously to Kato’s construction of theta functions as norm invariant
elements with a given degree zero divisor [Kat] (or 1 − z as as the trace-fixed unit on A1 − {1}
(0)
with divisor 1), we start with a degree 0 formal sum ec ∈ K0 of c-torsion points on E × E ′ that
is fixed by the action of the subgroup Γ of GL2 (F ) of automorphisms of E × E ′ . Pulling back
(0) (0)
by the resulting map Z → K0 gives an extension of Z by K2 that is the class of a 1-cocycle

c ΘE×E ′ : Γ → K2 (L(E × E ′ )) ⊗ Z[ 30
1
].

The particular choice of cocycle is given by a certain sum of theta functions on E and E ′
pulled back to divisors on E × E ′ with residue ec .
Now, unless the class number of F is 1, there is no action of arbitrary Hecke operators
on the first Γ-cohomology of K2 (L(E × E ′ )). Rather, we define an action on the collection
of such cohomology groups as E and E ′ run over the isomorphism classes of elliptic curves
with CM by O, noting here that Γ varies with E × E ′ . That is, in Section 2 we define the
notion of a ∆-module system and Hecke operators on the cohomology group of such a system,
which we employ for this purpose. In Proposition 3.3.3, we show that the tuple c Θ of classes
is Eisenstein with respect to Hecke operators attached to prime ideals.
Though representative elliptic curves can be defined over the Hilbert class field of F , the
torsion on such a curve does not always define an abelian extension of F . Fortunately, as in
[deS], we may choose representative elliptic curves with this property over any ray class field
L of level f such that O× injects into (O/f)× . We restrict a slight modification of c ΘE×E ′ to a
congruence subgroup of Γ that one might typically denote Γ0 (N ), for an O-ideal N prime to
c. This takes values in a direct limit of motivic cohomology groups H 2 (U, 2), where U is an
open in E × E ′ containing all points (0, Q) with Q a primitive N -torsion point on E ′ . We can
then pull back using a good choice of Q to obtain a specialized cocycle
1
c ΘN ,E×E ′ : Γ0 (N ) → K2 (F (N ∩ f)) ⊗ Z[ 30 ].

In Proposition 4.1.4, we show that by varying the f implicit in the definition, we obtain a
cocycle valued in K2 (F (N )) ⊗ Z[ 301
]. The tuple c ΘN of classes as we vary E × E ′ remains
Eisenstein away from the level. In Proposition 4.2.4, we show that this cocycle takes values in
K2 (O′ (N )) ⊗ Z[ 30
1
] using motivic cohomology over Dedekind schemes.
We also treat the dependence of our tuples of classes on the auxiliary ideal c in Corollary
4.1.9, which requires a subtle analysis of pullbacks of our cocycles. As with elliptic units that

4
are the pullbacks of theta functions at torsion points, one might hope that there exists a tuple
ΘN with (N c2 − R(c)) ◦ ΘN = c ΘN up to some controllable denominators. Here, of course,
the consideration of denominators is complicated by the fact that we are working with the finite
group K2 (O′ (N )). In Theorem 4.3.3, we show that ΘN exists upon tensor product with Zp
and the taking of an eigenspace for a character χ of the prime-to-p part of (O/N )× , avoiding
one particular choice of χ.
Next, we note that the restrictions of our cocycles to Γ1 (N )-type subgroups are homomor-
phisms, as the action of Γ0 (N ) on K2 (F (N )) is through the Artin reciprocity map applied to
the lower right-hand corner of such a matrix. We note that there are h = |Cl(F )| isomorphism
classes of elliptic curves with CM by O, and our tuple c Θ is of h2 cocycles corresponding
to the pairs of representatives of these classes. On the other hand, the Bianchi space Y1 (N )
1
of level N has h connected components. To define a map on H1 (Y1 (N ), Z[ 30 ]), we take E
∼ ′
to the the representative curve satisfying E(C) = C/O and vary E . We then obtain the
Eisenstein map c ΠN in the theorem (see Section 5.1). In doing so, we account for the fact that
the Hecke operators we defined, which are well-suited for describing the action on our tuple
of h2 cocycles, do not agree with the usual Hecke operators on cohomology.
Unlike the cocycles of [ShVe], we do not expect the cocycles defining c Θ to be parabolic,
which is to say that they are not coboundaries on parabolic subgroups. However, we do expect
parabolicity to hold for the specialization c ΘN . In Proposition 5.2.8, we prove a parabolicity
result for the resulting maps on homology by comparison of the eigenvalues of our Hecke
operators on c ΠN with the eigenvalues of Hecke operators on the homology of the boundary of
the Borel-Serre compactification of Y1 (N ). In this setting, parabolicity means that the maps
c ΠN on the homology of Y1 (N ) factor through the homology of the Satake compactification
X1 (N ). Putting this together with our independence from c result, we obtain in Theorem 5.2.9
a map ΠχN on the χ-eigenspace of H1 (X1 (N ), Zp ), where N is divisible by at most one power
of each prime over p and χ is a character on (O/N )× not on a certain short list.
Ideally, we would have an Eisenstein map Π◦N defined on the homology of X1 (N ) relative
to certain cusps, taking Manin-type symbols (cf. Cremona’s work [Cre]) to Steinberg symbols
of elliptic units. There are several obstacles, not least that our proof that ΠN exists in some
cases as a map on the homology of the compactification X1 (N ) is indirect and does not
follow from a statement about the tuple c Θ. On the other hand, there are no evident Steinberg
relations among general elliptic N -units, so no obvious way in which to define such a map
on the larger relative homology group directly. Moreover, Cremona’s symbols are defined
only in the Euclidean setting, and most generalizations don’t seem ideally suited to such a
treatment. Additionally, the elliptic units we would want to consider are not in general true
elements of a unit group, but rather roots thereof, and thus need in general to be modified by

5
some auxiliary ideal c that complicates the derivation of formulas in our approach even in the
Euclidean setting. Thus, we have omitted any treatment of such formulas in this paper, despite
the fact that the connection with Steinberg symbols of elliptic units is almost implicit in our
constructions. We have some novel ideas for overcoming many of the obstacles we have just
described, but this is left for future work.
Acknowledgments. The first author completed most of this project at the Yau Mathematical
Sciences Center of Tsinghua University and was also funded by the Beijing Institute of Math-
ematical Sciences and Applications. The final details of this project were completed at the
Institute for Theoretical Sciences of Westlake University. He was also supported by the Insti-
tute for Advanced Study in Princeton during Spring 2022 and funded by the National Natural
Science Foundation of China under Grant No. 12050410242.
The second author thanks Akshay Venkatesh for his insights during their joint work and
Takako Fukaya and Kazuya Kato for early conversations on this subject. He also thanks
everyone who encouraged him to explore these maps over many years, including Adebisi
Agboola and Cristian Popescu. This material was based in part upon work of the second
author supported by the National Science Foundation (NSF) under Grant No. DMS-2101889.
Part of this research was performed while the second author was visiting the Simons Laufer
Mathematical Sciences Institute, which is supported by NSF Grant No. DMS-1928930.
The third author would like to thank Adel Betina and Ming-Lun Hsieh for helpful discus-
sions.
The fourth author would like to express gratitude to Taiwang Deng, Yangyu Fan, and Yichao
Zhang for helpful discussions and insights regarding Bianchi modular forms and Eisenstein
cohomology. The fourth author is supported by the National Natural Science Foundation of
China under Grant No. 12331004.

2 Hecke actions on cohomology

2.1 Module systems
Let F be a number field and O be its ring of integers. For any nonzero ideal N of O, we
denote by ClN (F ) the ray class group of modulus N of F . Let h be the order of the class
group Cl(F ) of F . Set I = {1, . . . , h}, and fix representative O-ideals ar for r ∈ I of Cl(F )
with a1 = O. For each pair (r, s) ∈ I 2 , set ar,s = as a−1
r for brevity. Fix n ≥ 1.
Let R denote the profinite completion of O. For each r ∈ I, fix a finite idèle αr ∈ R
representing ar . Set G = GLn (AfF ) and fix an open subgroup U of the profinite group GLn (R).

6
 ˜ be a submonoid of G ∩ Mn (R) containing U . For i = (i1 , . . . , in ) ∈ I n , set
Let ∆

xi = diag(αi1 , . . . , αin ).
˜ i,j = x−1 ∆x
Given also j ∈ I n , set ∆ ˜ j . We then set ∆
˜i = ∆˜ i,i and Ui = x−1 U xi ⊂ ∆ ˜ i.
i i
Now set G = GLn (F ). For any i = (i1 , . . . , in ) and j = (j1 , . . . , jn ) ∈ I n , let ∆i,j =
˜ i,j ∩ G. In the case that ∆
∆ ˜ = G ∩ Mn (R), we have

∆i,j = {(au,v )u,v ∈ G | au,v ∈ aiu ,jv for all 1 ≤ u, v ≤ n},

independent of our choice of idèles. Note that for k ∈ I n , we have ∆i,j ∆j,k ⊆ ∆i,k . Set ∆i =
∆i,i and Γi = Ui ∩ G. The group Γi is commensurable with GLn (O), so its commensurator in
G is G by [Shi, Lemma 3.10]. For i = (1, . . . , 1), we have that ∆i ⊆ Mn (O) ∩ GLn (F ) and
Γi = GLn (O) ∩ U .
Definition 2.1.1.
a. A ∆-module system A indexed by J ⊆ I n is a collection of abelian groups Ai for i ∈ J
such that each g ∈ ∆i,j with i, j ∈ J provides a homomorphism g : Aj → Ai , which
satisfy

i. if g ∈ ∆i,j and g ′ ∈ ∆j,k for i, j, k ∈ J, then g ◦ g ′ = gg ′ : Ak → Ai , and

ii. the identity matrix 1n provides the identity homomorphism on Ai for each i ∈ J.

b. A homomorphism ϕ : A → B of ∆-module systems indexed by J is a collection of

homomorphisms ϕi : Ai → Bi for i ∈ J such that ϕi ◦ g = g ◦ ϕj : Aj → Bi for all
g ∈ ∆i,j .
In [RhWh], Rhie and Whaples defined a right action of an abstract Hecke ring on group
cohomology with coefficients in a module. Fixing a ∆-module system A indexed by some
J, this applies in particular to give a right action of the abstract Hecke algebra for the Hecke
pair (∆i , Γi ) on H q (Γi , Ai ) for all q ≥ 0 and i ∈ J. The Hecke operator that we define of an
element of ∆i,j for i, j ∈ J agrees with Rhie and Whaples’s operator for the inverse matrix in
the setting i = j where our constructions can be compared (i.e., they work with G-modules,
not ∆-module systems). Their action arises from an action on homogeneous cochains, while
we describe the corresponding action on inhomogeneous cochains.
Let us work somewhat more generally. We fix a second open subgroup U ′ of GLn (R)
contained in ∆ ˜ and then set Ui′ = x−1 U ′ xi and Γ′i = Ui′ ∩ G for i ∈ J. Given any g ∈ ∆i,j for
i
i, j ∈ J, we may decompose the double coset Γ′i gΓj as a finite union
v
a
Γ′i gΓj = gt Γj (2.1)
t=1

7
for some gt ∈ ∆i,j for 1 ≤ t ≤ v and some v ≥ 1.

Definition 2.1.2. Given a choice of double coset decomposition for g ∈ ∆i,j as in (2.1), we
define the Hecke operator of Γ′i gΓj on f : Γqj → Aj as T (g)f : (Γ′i )q → Ai given by
v
X
(T (g)f )(γ) = gσ(t) f (µt ),
t=1

where for γ = (γ1 , . . . , γq ) ∈ (Γ′i )q , the elements σ ∈ Sv and µt ∈ Γqj for 1 ≤ t ≤ v are
(q)
defined as follows: recursively setting ht = gt and
(w) (w−1)
γw ht = ht µt,w
(w)
with µt,w ∈ Γj and ht ∈ {g1 , . . . , gv } for 1 ≤ w ≤ q, we take µt = (µt,1 , . . . , µt,q ) and let
(0)
σ ∈ Sv be the unique permutation such that gσ(t) = ht for each 1 ≤ t ≤ v.

In general, the cochain T (g)f depends upon the set of representatives {g1 , . . . , gv } of
Γ′i gΓj .
However, fixing such a choice, one sees easily that T (g) defines a map of cochain
complexes, which also follows from the proof of the next result. This tells us that T (g) is
compatible with the connecting maps arising from short exact sequences of ∆-module systems.

Proposition 2.1.3. The above operation on cochains induces a homomorphism

T (g) : H q (Γj , Aj ) → H q (Γ′i , Ai )

that is independent of the choice of representatives in (2.1).

Proof. For all γ ∈ Γ′i and 1 ≤ t ≤ v, write

γgt = gσγ (t) τt (γ)

for some τt (γ) ∈ Γj and permutation σγ ∈ Sv . One can easily check that σγγ ′ = σγ σγ ′ and
τt (γγ ′ ) = τσγ ′ (t) (γ)τt (γ ′ ) for γ, γ ′ ∈ Γ′i .
For F ∈ HomZ[Γj ] (Z[Γq+1 j ], Aj ), we define T (g)F ∈ HomZ[Γ′i ] (Z[(Γ′i )q+1 ], Ai ) by
v
X
T (g)F (γ1 , . . . , γq+1 ) = gt F (τσγ−1 (t) (γ1 ), . . . , τσγ−1 (t) (γq+1 )) (2.2)
1 q+1
t=1

for g ∈ ∆i,j and γ1 , . . . , γq+1 ∈ Γ′i .

We compare T (g) with an operator

S(g) : HomZ[Γj ] (Z[Gq+1 ], Aj ) → HomZ[Γ′i ] (Z[Gq+1 ], Ai )

8
given on F ′ ∈ HomZ[Γj ] (Z[Gq+1 ], Aj ) by
v
X
′
S(g)F (δ1 , . . . , δq+1 ) = gt F ′ (gt−1 δ1 , . . . , gt−1 δq+1 )
t=1

for δ1 , . . . , δq+1 ∈ G, which clearly commutes with the standard differentials and is independent
of all choices.
Define
Πq : HomZ[Γj ] (Z[Γq+1
j ], Aj ) → HomZ[Γj ] (Z[Gq+1 ], Aj )
on F ∈ HomZ[Γj ] (Z[Γq+1
j ], Aj ) by Πq (F ) = F ◦ π q+1 , where π : G → Γj is given by

π(h) = h · s(Γj h)−1

for a chosen section s of the canonical surjection G → Γj \G that contains gt−1 for 1 ≤ t ≤ v
in its image. These (noncanonical) maps Z[Gq+1 ] → Z[Γq+1 j ] give a map of augmented
Z[Γj ]-projective resolutions of Z for the standard differentials, so Π· is a quasi-isomorphism.
Let F ∈ HomZ[Γj ] (Z[Γq+1
j ], Aj ), and let F ′ = Πq (F ). We have
v
X
S(g)F ′ (γ1 , . . . , γq+1 ) = gt F ′ (τσγ−1 (t) (γ1 )gσ−1
−1
(t)
, . . . , τσγ−1 −1
(t) (γq+1 )gσγ−1 (t)
)
1 γ1 q+1 q+1
t=1
v
X
= gt F (τσγ−1 (t) (γ1 ), . . . , τσγ−1 (t) (γq+1 )),
1 q+1
t=1

where the first equality follows by definition and the second since π(µgt−1 ) = µ for µ ∈ Γj
and 1 ≤ t ≤ v. Since this restriction also induces a quasi-isomorphism of complexes in the
opposite direction to Π· , it follows that T (g) is a map of complexes. Since S(g) is entirely
independent of choices, the maps on cohomology induced by T (g) are independent of choices
as well.
Given f : Γqj → Aj , consider the unique homogeneous cochain F as above such that

f (γ1 , . . . , γq ) = F (1, γ1 , . . . , γ1 · · · γq ),

and recall that this induces an isomorphism between the inhomogeneous and homogeneous

9
cochain complexes. The proposition now follows from the computation
v
X
T (g)F (1, γ1 , . . . , γ1 · · · γq ) = gt F (1, τσγ−1 (t) (γ1 ), . . . , τσγ−1 (t) (γ1 ) · · · τσγ−1q (t) (γq ))
1 1
t=1
Xv
= gt f (τσγ−1 (t) (γ1 ), . . . , τσγ−1···γq (t) (γq ))
1 1
t=1
Xv
= gσγ1 ···γq (t) f (τσγ2 ...γq (t) (γ1 ), . . . , τt (γq ))
t=1

= T (g)f (γ1 , . . . , γq ),

as σ = σγ1 ···γq and µt,w = τσγw+1 ···γq (t) (γw ) for 1 ≤ t ≤ v and 1 ≤ w ≤ q.

We have an alternative description of T (g), following along similar lines to [Hid, 9.4(c)].
For g ∈ ∆i,j , consider the operator

ϕg : H q (Γj ∩ g −1 Γ′i g, Aj ) → H q (gΓj g −1 ∩ Γ′i , Ai )

induced by the map taking a cochain f : (Γj ∩ g −1 Γ′i g)q → Aj to the cochain ϕg (f ) satisfying

ϕg (f )(γ1 , . . . , γq ) = gf (g −1 γ1 g, . . . , g −1 γq g)

for γ1 , . . . , γq ∈ gΓj g −1 ∩ Γ′i .

Proposition 2.1.4. The Hecke operator T (g) for g ∈ ∆i,j equals the composition
res ϕg cor
H q (Γj , Aj ) −→ H q (Γj ∩ g −1 Γ′i g, Aj ) −→ H q (gΓj g −1 ∩ Γ′i , Ai ) −→ H q (Γ′i , Ai ),

where res and cor denote restriction and corestriction, respectively.

Proof. Write
v
a
Γ′i = νt (gΓj g −1 ∩ Γ′i ).
t=1

with νt ∈ Γ′i for 1 ≤ t ≤ v. Then

v
a v
a
Γ′i gΓj = νt (gΓj g −1
∩ Γ′i )gΓj = νt gΓj ,
t=1 t=1

so setting gt = νt g ∈ ∆i,j , we have a coset decomposition as in (2.1). For γ ∈ Γ′i , let σγ ∈ Sv

and τt′ (γ) ∈ gΓj g −1 ∩ Γ′i for 1 ≤ t ≤ v be defined by γνt = νσγ (t) τt′ (γ). This implies that
γgt = gσγ (t) · g −1 τt′ (γ)g, which means that

τt (γ) = g −1 τt′ (γ)g ∈ Γj ∩ g −1 Γ′i g,

10
where τt (γ) is as in the proof of Proposition 2.1.3.
The corestriction map on homogeneous cochains sends C : (gΓj g −1 ∩ Γ′i )q+1 → Ai to
v
X
cor(C)(γ1 , . . . , γq+1 ) = νt C(τσ′ γ−1 (t) (γ1 ), . . . , τσ′ γ−1 (t)
(γq+1 ))
1 q+1
t=1

for γ1 , . . . , γq+1 ∈ Γ′i (cf. [NSW, Section 1.5]). Then, given a homogeneous cochain
F : Γq+1
j → Aj , we have
v
X
cor(ϕg (res(F )))(γ1 , . . . , γq+1 ) = νt ϕg (res(F ))(τσ′ γ−1 (t) (γ1 ), . . . , τσ′ γ−1 (t)
(γq+1 ))
1 q+1
t=1
v
X
= gt F (τσγ−1 (t) (γ1 ), . . . , τσγ−1 (t) (γq+1 ))
1 q+1
t=1

= T (g)F (γ1 , . . . , γq+1 ).

˜ with a certain property gives

We end this subsection by explaining how an element y ∈ ∆
rise to Hecke operators T (g) on any H q (Γj , Aj ).

Proposition 2.1.5. Let y ∈ ∆ ˜ be such that det(U ′ ∩yU y −1 ) = R× , and let i, j ∈ J be such that
det(x−1 ′ −1
i yxj ) has trivial ideal class. Then there exists g ∈ ∆i,j such that U xi gxj U = U yU ,
′

and the coset Γ′i gΓj is independent of the choice of g. In particular, T (g) provides a Hecke
operator
TA (y) : H q (Γj , Aj ) → H q (Γ′i , Ai )
depending only the double coset U ′ yU .

Proof. By strong approximation, det : G\G/V → (AfF )× /F × det(V ) is a bijection for any
open compact subgroup V of G. Of course, we have (AfF )× /F × R× = Cl(F ), so if det(V ) =
R× , then since det(x−1 i yxj ) is trivial in Cl(F ), there exist g ∈ G and v ∈ V such that
−1
g = xi yxj v. In particular, taking such a V contained in Uj , we have g ∈ ∆ ˜ i,j ∩ G = ∆i,j .
Further taking V = xj (y U y ∩U )xj so that v ∈ xj y U yxj , we then have xi gx−1
−1 −1 ′ −1 −1 ′ ′
j ∈ U y,
yielding the equality of double cosets.
Let g ∈ ∆i,j be as in the statement, which tells us that it satisfies Ui′ gUj = Ui′ x−1
i yxj Uj .
−1 −1
Set W = Ui ∩ gUj g , and note that W = xi u (U ∩ yU y )uxi for u ∈ U ′ such
′ −1 ′ −1

that g ∈ x−1 −1 ×
i u yxj Uj , so det(W ) = R by assumption. By strong approximation again,
G\G/Ui′ → G\G/W is a bijection, so (G ∩ Ui′ )\Ui′ /W is a singleton, which is to say that
Ui′ = Γ′i W . In particular, we have

Ui′ gUj = Γ′i g(g −1 W g)Uj = Γ′i gUj .

11
It follows that Ui′ x−1 ′ ′
i yxj Uj ∩ G = Γi gΓj , and therefore the latter double coset is independent
of g.

2.2 Hecke operators as correspondences

The description of Hecke operators given by Proposition 2.1.4 allows for comparison with
Hecke actions on locally symmetric spaces. Let A be a ∆-module system indexed by a set
Qn
J = {f (r) | r ∈ I} for a function f : I → I n such that each a−1 r u=1 af (r)u is principal (e.g.,
f (r) = (r, 1, . . . , 1)). In this subsection, we shall henceforth identify J with I so that Af (r) is
denoted more simply by Ar , and similarly with other subscripts, such as on Γ.
Let Hn,F denote the symmetric space GLn (F ⊗Q R)/(F ⊗Q R)× On (F ⊗Q R), where
On (F ⊗Q R) agrees with the product of degree n orthogonal and unitary groups of the
completions of F at its real and complex places, respectively. Suppose that det(U ) = R× , in
which case the locally symmetric space

Y (U ) = GLn (F )\(GLn (AfF ) × Hn,F )/U,

with G = GLn (F ) acting diagonally and U acting on G = GLn (AfF ) on the right, is a disjoint
union of components homeomorphic to Yr = Γr \Hn,F for r ∈ I, with the inclusion of Yr in
Y = Y (U ) given by z 7→ (x−1 r , z).
The coefficient system A = A(U ) on Y given by Ar = Ar (U ) = Γr \(Hn,F × Ar ) on Yr
gives rise to a constructible sheaf on Y (i.e., of its continuous sections) that we again give the
notation A, with its restriction to Yr denoted Ar .
For each r ∈ I, we suppose that the orders of all torsion elements in Γr act invertibly on
Ar and that the scalar elements in Γr act trivially on Ar . We then have isomorphisms

H q (Γr , Ar ) ∼
M
= H q (Y, A) (2.3)
r∈I

for q ≥ 0. These can be described as the sum over r of the compositions

∼ ∼
H q (Γr , Ar ) −
→ H q (Γr , H 0 (Hn,F , Ar )) −
→ H q (Yr , Ar )
∼
the first map coming from the canonical isomorphism Ar − → H 0 (Hn,F , Ar ). The inverse of
the second map is induced by the chain map taking an Ar -valued simplicial q-cochain F on Yr
to the cochain f : Γqr → Ar such that f (γ1 , . . . , γq ) is the function taking x ∈ Hn,F to the value
of F on the image in Yr of the geodesic q-simplex on Hn,F with vertices x, γ1 x, . . . , γ1 . . . γq x.
We define actions of Hecke operators on the left of (2.3). We focus here on the case
that U = U ′ for simplicity of notation, but distinct U and U ′ can be treated by making the

12
 g
necessary changes of notation. For g ∈ GLn (F ), set Yr,s = (Γr ∩ gΓs g −1 )\Hn,F , and consider
the coefficient system
Agr,s = (Γr ∩ gΓs g −1 )\(Hn,F × Ar ).
g g −1
Any g ∈ ∆r,s defines a homeomorphism g : Ys,r → Yr,s via left multiplication on Hn,F .
g −1
Together with the map g : As → Ar , this induces a map ϕg : H q (Ys,r , As ) → H q (Yr,s
g
, Ar ) on
sheaf cohomology.
Definition 2.2.1. For g ∈ ∆r,s , we define the Hecke operator T (g) : H q (Ys , As ) → H q (Yr , Ar )
of g by
res −1 ϕg cor
H q (Ys , As ) −→ H q (Ys,r
g
, As ) −→ H q (Yr,s
g
, Ar ) −→ H q (Yr , Ar ),
where here the maps res and cor are restriction and trace, respectively.
The following is then a consequence of Proposition 2.1.4.
Proposition 2.2.2. For g ∈ ∆r,s , the diagram
T (g)
H q (Γs , As ) H q (Γr , Ar )
≀ ≀
T (g)
H q (Ys , As ) H q (Yr , Ar )
commutes.
In particular, it follows from Proposition 2.1.3 that the Hecke operator T (g) : H q (Ys , As ) →
H q (Yr , Ar ) is independent of the choice of representatives of Γr gΓs .
We can also view these T (g) for g ∈ ∆r,s as coming from a single adelic operator.
Being a bit unrigorous in our notation for motivational purposes, it is useful to note that
ϕg (x−1 g
s , z, as ) ∈ Ar,s may be viewed in terms of representatives of double cosets as

(x−1 −1 −1 −1 −1 −1 −1 −1 −1
r , gz, gas ) = (g xr , z, as ) = (xs (xs g xr ), z, as ) = (xs , z, as ) · (xr gxs ) ,

where the latter multiplication is that of GLn (AfF ) on the right. We remark that xr gx−1 ˜
s ∈ ∆
by definition of ∆r,s .
For y ∈ G, let U y = U ∩ yU y −1 . Right multiplication by y −1 defines a homeomorphism
−1
y −1 : Y (U y ) → Y (U y ). It also gives rise to a map on cohomology, as we now explain.
˜ such that det(U y ) = R× , there exists a unique homomorphism
Proposition 2.2.3. For y ∈ ∆
−1
ψy : H q (Y (U y ), A) → H q (Y (U y ), A)

that, for each pair (r, s) ∈ I 2 such that the ideal attached to det(x−1
r yxs ) is principal, restricts
to ϕg for any g ∈ ∆r,s such that
U y xr gx−1
s = U y.
y
(2.4)

13
Proof. The existence of g ∈ G satisfying (2.4) is by strong approximation. Any such g lies in
∆r,s , which we can see by rearranging (2.4) as

g ∈ x−1 y −1 ˜ ˜
r U xr · xr yxs ⊆ Ur · ∆r,s = ∆r,s . (2.5)

The condition (2.4) implies the two equalities U y −1 xr = U xs g −1 and U yxs = U xr g. The
former gives
x−1 y
r U xr ∩ G = Γr ∩ gΓs g ,
−1
(2.6)
−1 −1
and the latter tells us that x−1 s U
y
xs ∩ G = Γs ∩ g −1 Γr g. Thus, H q (Y (U y ), A) has
g −1
a direct summand isomorphic to H q (Ys,r , As ), and H q (Y (U y ), A) has one isomorphic to
−1
H q (Yr,s
g
, Ar ). The groups H q (Y (U y ), A) and H q (Y (U y ), A) are the direct sums of these
summands over r ∈ I, so ψy exists. As for uniqueness, (2.5) and (2.6) together show that if
g, g ′ ∈ ∆r,s both satisfy (2.4), then g ′ g −1 ∈ Γr ∩ gΓs g −1 , so ϕg = ϕg′ .

Definition 2.2.4. For q ≥ 0, we define the adelic Hecke operator

TA (y) : H q (Y, A) → H q (Y, A)

of y ∈ ∆˜ such that det(U y ) = R× as the composition of pullback to Y (U y−1 ), the map ψy ,

and the trace from Y (U y ) to Y .

The following is a corollary of Proposition 2.2.3 with the observation from Proposition
2.1.5 that the coset Γr gΓs depends only on U yU .

Proposition 2.2.5. For y ∈ ∆ ˜ for such that det(U y ) = R× and s ∈ I, let r ∈ I be unique
such that the ideal attached to det(x−1
r yxs ) is principal. Let g ∈ ∆r,s be any element such that

U xr gx−1
s U = U yU.

Then the following diagram commutes

T (g)
H q (Ys , As ) H q (Yr , Ar )

TA (y)
H q (Y, A) H q (Y, A).

Putting Propositions 2.2.2 and 2.2.5 together with Proposition 2.1.5, we see that the con-
structions of TA (y) on the left and right-hand sides of (2.3) coincide.

14
2.3 Hecke operators attached to ideals
Let us suppose now that A = (Ai )i∈I n is a ∆-module system for I n . We impose two additional
conditions on our open subgroup U of GLn (R):

i. U is normalized by diagonal matrices in GLn (R), and

ii. U contains the subgroup of G consisting of diagonal matrices in GLn (O).

Condition (i) implies that the group Γi is independent of the choice of the αr representing ar
for r ∈ I. Condition (ii) allows us to make the following definition, independent of choice.

Definition 2.3.1. Let n1 , . . . , nn be nonzero ideals of O. For each i ∈ I n , let j ∈ I n be unique

such that aiu ,ju nu is principal for all 1 ≤ u ≤ n. Let ηu be a generator of aiu ,ju nu for each u,
and set g = diag(η1 , . . . , ηn ). If g ∈ ∆i,j , then we define the Hecke operator

T (n1 , . . . , nn ) : H q (Γj , Aj ) → H q (Γi , Ai )

for (n1 , . . . , nn ) as T (g).

We will be particularly interested in the following operators.

Definition 2.3.2. For a nonzero ideal n of O and 1 ≤ u ≤ n, we set

Tn(u) = T (n, . . . , n, 1, . . . , 1)

when the latter operator exists, the expression for which contains u copies of n. We write
(1) (n)
Tn = Tn and [n]∗ = Tn .

We have the following simple lemma regarding the operators [n]∗ .

Lemma 2.3.3. Let i, j ∈ I n be such that aiu ,ju n is principal for all 1 ≤ u ≤ n. If [n]∗ exists,
then its double coset decomposition is
 η1   η1 
Γi . .. Γj = . .. Γj ,
ηn ηn

where ηu is a generator of aiu ,ju n for each u.

Proof. Let h = diag(η1 , . . . , ηn ). The idèle ηu−1 αi−1 u

αiv ηv has associated ideal generating
(aiu ,ju n) aiu ,iv aiv ,jv n = aju ,jv for u, v ∈ {1, . . . , n}, so it is a multiple of αj−1
−1
u
αjv by a unit in
−1 −1
R. Therefore, h−1 Ui h = h−1 xi U xi h is conjugate to Uj = xj U xj by a diagonal matrix in
GLn (R), so it is equal to Uj by assumption on U . Then h−1 Γi h = Γj , so we are done.

15
 For an ideal a of O, let Fa× denote the group of finite idèles of F that are 1 at all primes not
dividing a. For r ∈ I, we now choose the idèle αr ∈ R with associated ideal ar to lie in Fa×r .
Let N be a nonzero ideal of O that is prime to ar for r ∈ I. Let U (N ) denote the open
subgroup of GLn (R) consisting of matrices with image in GLn (O/N ) contained in the image
of the diagonal matrices in GLn (O). If U contains U (N ), we refer to the largest ideal M such
that U contains U (M) as the level of U . We suppose that U has level N .
Let us define an adelic analogue of Definition 2.3.1 that works for more pairs of elements
of I n but provides slightly different Hecke operators in cases where both are defined.

Definition 2.3.4. Let n1 , . . . , nn be nonzero ideals of O. Let νu ∈ Fn×u have associated ideal
nu . For all primes p of O dividing nu + N , we assume that U contains all diagonal matrices
in GLn (Rp ) with vth entry 1 for all v ̸= u. For i, j ∈ I n such that nu=1 aiu ,ju nu is principal,
Q

we define
Ti,j (n1 , . . . , nn ) : H q (Γj , Aj ) → H q (Γi , Ai )
to be the restriction of TA (diag(ν1 , . . . , νn )).

If we suppose that the orders of torsion elements in Γi acts invertibly and scalar elements
act trivially on Ai for each i ∈ I n , then we can also define Ti,j (n1 , . . . , nn ) : H q (Yj , Aj ) →
H q (Yi , Ai ) as the restriction of TA (diag(ν1 , . . . , νn )), and these operators are compatible in
the sense of Proposition 2.2.2.
Remark 2.3.5. Suppose that the ideals n1 , . . . , nn are prime to N . Taking i, j ∈ I n such that
aiu ,ju nu is principal for all 1 ≤ u ≤ n, the operator Ti,j (n1 , . . . , nn ) is given by T (g) with g
diagonal and congruent to the identity matrix modulo N , whereas T (n1 , . . . , nn ) is T (h) for h
the diagonal matrix with uth entry a generator of aiu ,ju nu , which may not be 1 modulo N .
We are concerned in this paper in a situation for which n = 2. For the rest of this
subsection, take ∆˜ to be the submonoid ∆˜ 0 (N ) of GL2 (Af ) ∩ M2 (R) consisting of elements
F
with bottom row (0, z) modulo N , where z is prime to N . We shall need the following two
open subgroups of GL2 (R) of level N contained in ∆ ˜ 0 (N ). The first is the group U0 (N )
consisting of matrices with second row congruent to (0, a) modulo N for some a ∈ R× . The
second is the subgroup U1 (N ) consisting of matrices in U0 (N ) with (2, 2)-entry congruent to
an element of O× modulo N . We set Γ∗ (N )i = U∗ (N )i ∩ G for ∗ ∈ {0, 1}. For the remainder
of this subsection, we take n = 2 and suppose that U contains U1 (N ).

Definition 2.3.6. Let d be an ideal of O prime to N . Let i, j ∈ I 2 be such that ai1 ,j1 d−1 and
ai2 ,j2 d are principal. Let λ be a generator of the latter ideal. For any matrix δ ∈ ∆0 (N )i,j with

ai1 ,j1 ai2 ,j2 = det(δ)O

16
and bottom right entry λ modulo N , we define the (adjoint) diamond operator ⟨d⟩∗ attached
to d by
⟨d⟩∗ = T (δ) : H q (Γj , Aj ) → H q (Γi , Ai )
for q ≥ 0. Note that such a matrix δ exists by the strong approximation theorem.

The reader may verify the following.

Lemma 2.3.7. We maintain the notation of Definition 2.3.6.

a. We have Γi δΓj = δΓj , and δΓj depends only upon the image of d in ClN (F ).

b. We have δ −1 ∈ ∆0 (N )j,i , and T (δ −1 ) = ⟨e⟩∗ for any ideal e of O inverse to d in ClN (F ).

The following amounts to a special case of Remark 2.3.5.

Lemma 2.3.8. Let n be a nonzero ideal of O. Let q ≥ 0.

a. Suppose that n is prime to N . For i, j, k ∈ I 2 such that ai1 ,k1 n, ai2 ,k2 n, and ai1 ,j1 n−1 are
principal and j2 = k2 , we have

[n]∗ = ⟨n⟩∗ ◦ Tj,k (n, n) : H q (Γk , Ak ) → H q (Γi , Ai ).

b. For i, j ∈ I 2 such that ai1 ,j1 n is principal and i2 = j2 , we have

Tn = Ti,j (n, 1) : H q (Γj , Aj ) → H q (Γi , Ai ).

Proof. Let ηu generate aiu ,ku n for u ∈ {1, 2}, and let g = ( η1 η2 ). Let ν ∈ R× with
ν ≡ η2 mod N . Let δ̃ ∈ ∆ ˜ i,j be diagonal with (1, 1)-entry αj1 α−1 ν −1 and (2, 2)-entry
i1
−1 ∗
αj2 αi2 ν. Then Ui δ̃Uj = Ui δUj for δ ∈ ∆i,j with T (δ) = ⟨n⟩ , with both double cosets equal
to a single left coset.
The matrix δ̃ −1 g ∈ ∆˜ j,k has first and second diagonal entries with associated ideals aj1 ,k1 n
and n, respectively, with the second being 1 modulo N . Since U contains U1 (N ), it contains
all diagonal matrices with (1, 1)-entry in R× and (2, 2)-entry in O× modulo N . It follows that
the double coset Uj δ −1 gUk = Uj δ̃ −1 gUk is equal to the double coset of an element of ∆˜ j,k that
yields the operator Tj,k (n, n) = T (δ −1 g) of Definition 2.3.4. This yields part (a).
Part (b) is essentially immediate, again using the fact that U contains U1 (N ).

17
2.4 ∆-module systems with pushforwards
In our applications, ∆-module systems are direct limits of motivic cohomology groups of open
subschemes of commutative group schemes, with the elements of ∆ (or more precisely, each
∆i,j ) providing isogenies between these group schemes. The ∆-action on A is then one of
pullback. However, these motivic cohomology groups also come equipped with pushforwards
by elements of ∆, which we shall have occasion to employ. To fit this into our abstract
framework, let us define the notion of pushfoward maps on a ∆-module system. To distinguish
these from the maps in a ∆-module system, we use the standard notation for pullbacks for the
latter maps, writing g : Aj → Ai for g ∈ ∆i,j as g ∗ in this subsection.

˜ ′ be a submonoid of ∆.
Definition 2.4.1. Let A be a ∆-module system for J ⊆ I n , and let ∆ ˜
A system of ∆′ -pushforwards on A is a collection of pushforward maps g∗ : Ai → Aj with
g ∈ ∆′i,j for i, j ∈ J such that

i. h∗ ◦ g∗ = (gh)∗ if h ∈ ∆′j,k ,

ii. (1n )∗ is the identity on Ai , and

iii. g∗ ◦ g ∗ = (det(g)1n )∗ .

We have included (iii) as a standard compatibility of pushforwards and pullbacks, but we

do not actually use it in this paper. In fact, we eschew presenting any semblance of a general
theory and focus only on the single definition needed in this work.
Let us suppose that U is an open subgroup of GLn (R) satisfying conditions (i) and (ii) of
Section 2.3. Let A be a ∆-module system with a system of pushforwards for the submonoid
of diagonal matrices in ∆.˜

Definition 2.4.2. Given i, j ∈ I n and a nonzero ideal b of O, we write i ∼b j to denote that

aiu ,ju b is principal for all 1 ≤ u ≤ n.

Definition 2.4.3. Let b be a nonzero ideal of O. Let i, j ∈ J be such that i ∼b j. Let ρ ∈ ∆i,j
be diagonal with uth entry generating aiu ,ju b. For f : Γqi → Ai we define [b]∗ (f ) : Γqj → Aj
on µ ∈ Γqj by
[b]∗ (f )(µ) = ρ∗ f (ρµρ−1 ).

If b = (b) is principal, then we also write [b]∗ for [b]∗ . As Γj contains the diagonal matrices
in GLn (O), the following is easily verified.

18
Lemma 2.4.4. Let i, j ∈ J and b be a nonzero ideal of O such that i ∼b j. Suppose that every
choice of ρ∗ as in Definition 2.4.3 has the property that

ρ∗ γ ∗ = (ρ−1 γρ)∗ ρ∗ (2.7)

for all γ ∈ Γi . Then [b]∗ defines a homomorphism of chain complexes, and it depends upon
the choice of ρ only up to chain homotopy. Given a homomorphism ϕ : A → B of ∆-module
systems, we have [b]∗ ◦ ϕi = ϕj ◦ [b]∗ for any fixed choice of ρ.

3 GL2-cocycles for CM elliptic curves

3.1 CM elliptic curves
Keeping the notation of the last section, we now let F be an imaginary quadratic field. All
number fields will be considered as subfields of the algebraic numbers in C. There are
h = |Cl(F )| isomorphism classes of elliptic curves over C with CM by O. Each has a
representative defined over the Hilbert class field H of F . Recall that I = {1, . . . , h}.
We give a quick proof of the following known lemma.

Lemma 3.1.1. Suppose that L is a finite extension of F . Let E and E ′ be L-isogenous elliptic
curves with CM by O defined over L. Then all isogenies from E to E ′ are defined over L.

Proof. By [Sil, Theorem II.2.2], this is true for E = E ′ . Thus HomL (E, E ′ ) is a nonzero
O-submodule of Hom(E, E ′ ), hence of finite index. For f ∈ Hom(E, E ′ ), let m ≥ 1 be such
that mf = f ◦ m is defined over L. Since multiplication by m is also defined over L, we have
f ◦ m = f σ ◦ m for any automorphism of C fixing L. Then m(f − f σ ) = 0, so f = f σ .

Let us fix a nonzero ideal f of O prime to all ar for r ∈ I such that O× → (O/f)× is
injective. That is, we suppose that no nontrivial root of unity in O is 1 modulo f.
As described in the discussion of [deS, 1.4], there exists an elliptic curve over L = F (f)
with CM by O such that its torsion points are all defined over an abelian extension of F . In fact,
we may choose such a curve E1 so that E1 (C) ∼ = C/O and fix such an analytic isomorphism.
Let N be a nonzero ideal of O also prime to ar for all r ∈ I. Let R denote the Artin
σr
map for F (f ∩ N )/F , and let σr = R(a−1 r ) for r ∈ I. Set Er = E1 . By Lemma 3.1.1,
all isogenies between the curves Er are defined over L as well. In fact, by CM theory (cf.
[deS, 1.5]), there is a unique L-isogeny ψr : Er → E1 with kernel Er [ar ] that agrees with
σr−1 on prime-to-ar -torsion. The identification of E1 with Er /Er [ar ] gives rise to an analytic
isomorphism Er (C) ∼ = C/ar . In turn, this supplies an isomorphism HomL (Er , E1 ) ∼ = a−1
r .

19
For r, s ∈ I, since every analytic map C/ar → C/as preserving 0 corresponds to an isogeny
which is necessarily defined over L, we then have identifications

HomL (Er , Es ) ∼
= ar,s . (3.1)

Now, let us turn to the N -torsion on these elliptic curves. For α ∈ F × prime to N and
r ∈ I, let [α]r : Er [N ] → Er [N ] be the isomorphism given by multiplication by any element
in O congruent to α modulo N . Let us set σr,s = σs σr−1 = R(as,r ) for brevity.

Proposition 3.1.2. Let d ∈ ar,s be prime to N . We have

σr,s = d ◦ [d]−1 −1
r = [d]s ◦ d

as group homomorphisms Er [N ] → Es [N ].
−1
Proof. It suffices to prove this in the case s = 1. Observe that the quantity d ◦ [d]−1
r = [d]1 ◦ d
is independent of the choice of d ∈ a−1 −1
r . The isogeny ψr : Er → E1 is identified with 1 ∈ ar ,
and clearly 1 = 1 ◦ [1]−1 −1
r : Er [N ] → E1 [N ]. Since ψr equals σr on Er [N ], we are done.

Let t ∈ I be such that Et (C) ∼ = C/N . A fixed generator of N a−1 t (unique up to unit)
provides an isomorphism C/at → C/N of elliptic curves, and this gives an isomorphism
Et (C) ∼= C/N . We let Q be the primitive N -torsion point of Et corresponding to 1 ∈ C/N .
Finally, we set Pr = σt,r (Q) ∈ Er [N ] for all r ∈ I. Proposition 3.1.2 may then be rephrased
as follows.

Corollary 3.1.3. Let d ∈ ar,s be prime to N . Then

Ps = [d]−1
s (d · Pr ).

We will also have use of the following.

Lemma 3.1.4. Let d ∈ O be prime to N with d ≡ 1 mod f. For r ∈ I, we have

R(d)(Pr ) = [d]r (Pr ).

Proof. By CM theory (see [deS, Proposition 1.5]), we know that there exists an isogeny
λ : Er → Er with kernel Er [d] that agrees with R(d) on prime-to-d-torsion in Er , and R(d)
fixes Er [f]. The only endomorphism of Er with kernel Er [d] and fixing Er [f] is multiplication
by d, since there are no nontrivial units in O× that are 1 modulo f.

20
3.2 Motivic complexes of products of elliptic curves
Let R be Z or an order in the ring of integers of an imaginary quadratic field. For j ∈ {1, 2},
let Aj be an elliptic curve with endomorphism ring R over a characteristic 0 field L, and set
A = A1 ×L A2 . As in [ShVe, (2.5)], we have a complex K(A) in homological degrees 2 to 0:
∂ ∂
M M
K2 (L(A)) →− K1 (L(D)) → − K0 (L(x)),
D x

where Kd denotes the dth K-group, with the sums in degree d taken over the irreducible
L-cycles of dimension d in A. The homology of this complex is given by

Hd (K(A)) ∼
= H 4−d (A, Z(2)).

We have trace (i.e., pushforward) maps [α]∗ for multiplication by elements α ∈ R − {0}
on these groups. As such, we introduce the following notation. Let Z′ = Z[ 30 1
] throughout.
For an abelian group B, we set BZ′ = B ⊗ Z′ . For an abelian group M with a multiplicative
(R − {0})-action, we use M (0) to denote the subgroup of elements of MZ′ that are fixed under
all elements of R prime to some nonzero element. We refer to M (0) as the trace-fixed part of
M.

Lemma 3.2.1. The trace-fixed part of K0 (A) satisfies

K0 (A)(0) ∼
= lim H 0 (A[n], Z)(0) ,
−→
n

where n runs over the nonzero ideals of R. In fact, the class of A[n] is trace-fixed and is a sum
of distinct trace-fixed classes that generate H 0 (A[n], Z)(0) .

Proof. Any irreducible zero-cycle containing a non-torsion point clearly cannot be fixed under
multiplication by any nonunit in R − {0}. On the other hand, A[n] is fixed by all elements
of R − {0} prime to n. It is a sum of irreducible cycles that freely generate H 0 (A[n], Z).
The sum of all (R − {0})-multiples of such an irreducible cycle over elements prime to n
provides a trace-fixed class, from which the result is clear (and did not require working with
Z′ -coefficients).

It follows from [ShVe, Proposition 6.1.2] that H i (A, Z(2))(0) = 0 for all i ̸= 4, and

H 4 (A, Z(2))(0) ∼
= Z′ .

In fact, this is already true for (Z − {0})-fixed parts. The resulting surjection K0 (A)(0) → Z′
takes the class of a trace-fixed cycle to the order of its group of C-points. We call this map the
degree map.

21
Lemma 3.2.2. The image of the residue map K1 (A)(0) → K0 (A)(0) is the kernel of the degree
map.

Proof. Let C be the span under prime-to-n multiplication maps of a connected component of
A2 [n] for positive integer n, which we refer to as a component in this proof. (We eschew any
analysis of these, as it is not required for our purposes.) As in [Kat, 1.10] and [ShVe, (6.5)], if
we consider the scheme A1 × C (omitting the subscript L on the product), then since the cycle
C is fixed by prime-to-n multiplication, we have an exact sequence

0 → H 1 ((A1 − A1 [n]) × C, 1)(0) → H 0 (A1 [n] × C, 0)(0) → Z′ , (3.2)

with the final map the degree map. This forms a subcomplex of K(A)(0) → Z′ .
The components of A[n] have the form C × D, where C is a component of A1 [n] and the
D is a component of A2 [n]. The degree of C × D is the product of the degrees of C and D.
It suffices to see that every element of degree zero in H 0 (A[n], 0)(0) is a sum of elements of
degree zero in H 0 (C × A2 [n], 0)(0) and H 0 (A1 [n] × D, 0)(0) for some C and D. For this, note
that the classes of {0} in A1 [n] and A2 [n] have degree one.
Suppose we give each row and column of a matrix of a certain size a fixed positive integral
weight, with the first row and column having weight 1, and we define the weight of an entry
as the product of these. The result then amounts to the fact any such integral matrix with
weighted sum of its entries equal to zero is a sum of two matrices, one in which each row
has weighted sum zero and one in which each column does. In fact, one can choose the latter
matrix to be zero outside of the first column.

We also have the following.

Proposition 3.2.3. The sequence

0 → K2 (A)(0) → K1 (A)(0) → K0 (A)(0) → Z′ → 0

is exact.

Proof. Left exactness is [ShVe, Lemma 6.2.1], surjectivity of the degree map holds as the
class of 0 ∈ A has degree one, and exactness at K0 (A)(0) is Lemma 3.2.2.

3.3 Eisenstein cocycles for products of CM curves

Let us return to our situation of interest, using the notation of Section 3.1. For i = (i1 , i2 ) ∈ I 2 ,
set
Ei = Ei1 ×L Ei2 .

22
Elements (au,v )u,v of the monoid ∆i,j give rise to morphisms Ei → Ej of abelian L-schemes
for i, j ∈ I 2 via the maps au,v : Eiu → Ejv . We set E = i∈I 2 Ei for convenience.
`

Let us set K(i) = K(Ei ). Pullback by elements of ∆i,j provides morphisms of complexes
K(j) to K(i), compatible with composition, which is to say that the K(i) form a complex of
∆-module systems. The complexes K(i)(0) are still of ∆-module systems, as the diagram
giving the two compositions Ei → Ej of g ∈ ∆i,j with multiplication by an α ∈ O that is prime
to the ideal attached to det(xi gx−1j ) is cartesian, as in the proof of [ShVe, Lemma 6.3.1].
With the notation of Section 2.1, we take U = GL2 (R) and ∆ ˜ = M2 (R) ∩ GL2 (Af ), so
F
Γi consists of the elements of ∆i with determinant in O× . As a consequence of Proposition
3.2.3, we have the connecting map

d(i) : ker(K0 (i)(0) → Z′ )Γi → H 1 (Γi , K2 (i)(0) )

in Γi -cohomology. Since the differentials in the complex K(i) are ∆i,j -compatible, d(i) and
d(j) are compatible with the Hecke operator T (g), as noted in Section 2.1.
Let (0) ∈ H 0 (E, 0)(0) denote the sum of the classes of 0 ∈ Ei over i ∈ I 2 . For a nonzero
ideal c of O, we set
M
2
c e = (c ei )i∈I 2 = N c (0) − E[c] ∈ ker(K0 (i)(0) → Z′ )Γi .
i∈I 2

Then the direct sum d of the d(i) applied to c e is a class

M
c Θ = (c Θi )i∈I 2 ∈ H 1 (Γi , K2 (i)(0) ).
i∈I 2

If c = O, then this class is zero.

Lemma 3.3.1. Let i, j ∈ I 2 and n be an ideal of O prime to c such that i ∼n j. Then

[n]∗ (c Θi ) = c Θj (3.3)

in H 1 (Γj , K2 (j)(0) )

Proof. Since n is prime to c, we have [n]∗ (c ei ) = c ej . The base change condition in (2.7)
holds on motivic cohomology as ρ is proper and γ is flat (in fact, an isomorphism), and the
commutative square given by the two compositions γ ◦ ρ = ρ ◦ (ρ−1 γρ) is cartesian. The
lemma then follows by Lemma 2.4.4.

For any nonzero ideal n of O, we can view Tn and [n]∗ as acting on i∈I 2 H q (Γi , Kp (i)(0) )
L

for any q ≥ 0 and 0 ≤ p ≤ 2, and these operators act compatibly with residues.

23
Lemma 3.3.2. Let c be a nonzero ideal of O, and let p be a prime ideal of O. Then
[p]∗ E[c] = E[pc], and Tp − (N p + [p]∗ ) annihilates E[c].
Proof. Let i, j ∈ I 2 be such that paiu ,ju is principal, and let ηu ∈ F be a generator, for
u ∈ {1, 2}. We can then view ηu as an element of HomL (Eiu , Eju ) under its identification
with aiu ,ju from (3.1). The pullback of Ej [c] by ( η1 η2 ) is then Ei [pc], so [p]∗ E[c] = E[pc].
Now let i, j ∈ I 2 with i2 = j2 and pai1 ,j1 = (η1 ). Since Tp = T (g) for g = ( η1 1 ), and g
commutes with the diagonal matrix defining [c]∗ , it suffices to consider the case c = O. The
right action of Γi on the pullback Ei1 [p] × {0} of 0 ∈ Ei by g factors through the quotient of
Γi by Γi ∩ (1 + p∆i ). This gives a compatible action of Ui through the isomorphic quotient of
x−1 −1 1 a
i GL2 (Rp )xi . Under the resulting pullback action, the matrices xi ( 0 1 ) xi with a running
through representatives of O/p, together with x−1 0 1
i ( 1 0 ) xi , carry Ei1 [p] × {0} to the N p + 1
distinct O-submodule schemes of Ei [p] isomorphic to O/p. Since Tp = TA (( π0 01 )) for π a
uniformizer of Rp and ( π0 a1 ) = ( 10 a1 ) ( π0 01 ) for a as above, while ( 10 π0 ) = ( 01 10 ) ( π0 01 ) ( 01 10 ),
the sum of the classes of these subschemes gives exactly the result of the Hecke action on the
class of 0 in Ej . We conclude that Tp (0) = N p(0) + E[p], as desired.
Proposition 3.3.3. For any prime ideal p and nonzero ideal c of O, we have Tp (c Θ) =
(N p + [p]∗ )c Θ.
Proof. By Lemma 3.3.2, we have Tp (c e) = (N p + [p]∗ )c e. The result then follows from the
Hecke-equivariance of the sum d of differentials.

We can also describe the action of diamond operators.

Lemma 3.3.4. Let d be an ideal of O prime to N . Then the diamond operator ⟨d⟩∗ fixes
(0) ∈ H 0 (E, 0).
Proof. Recall that δ ∈ ∆i,j with ⟨d⟩∗ = T (δ) has determinant generating ai1 ,j1 ai2 ,j2 , and
δ −1 ∈ ∆j,i by Lemma 2.3.7(b). In particular, if (x, y) ∈ C × C is such that (x, y)δ ∈ aj1 × aj2 ,
then (x, y) ∈ ai1 × ai2 . Therefore, pullback by δ on H 0 (Ej , 0) takes the class of 0 to the class
of 0 in H 0 (Ei , 0).
Corollary 3.3.5. Let d be an ideal of O prime to N , and let c be a nonzero ideal of O. Let
i, j ∈ I 2 be such that ai1 ,j1 d−1 and ai2 ,j2 d are principal. Then ⟨d⟩∗ c Θj = c Θi .
Lemma 3.3.6. Let i, j ∈ I 2 and n be a nonzero ideal of O such that i ∼n j. Let c be a nonzero
ideal of O. Then we have
[n]∗ (c Θj ) = cn Θi − N c2 · n Θi
in H 1 (Γi , K2 (i)).
Proof. This is immediate from the fact that [n]∗ (c ej ) = cn ei − N c2 n ei .

24
3.4 A representative cocycle
For i ∈ I 2 , let us define K2,N (i) as the trace-fixed part of the direct limit of groups H 2 (V, 2)
running over open subschemes V of Ei which contain all (O/N )× -multiples of (0, Pi2 ).
As ∆0 (N )i preserves the latter set, (K2,N (i))i∈I 2 is a ∆0 (N )-module system, and so is the
collection of trace-fixed parts. To pull back by (0, Pi2 ), we need a representative of the class
c Θi that takes values in K2,N (i).
Let c be a ideal of O prime to N . Recall that for r ∈ I, there exists a unique trace-fixed
element c θr ∈ H 1 (Er − Er [c], Z′ (1)) with boundary N c(0) − Er [c] (again, see [Kat, 1.10] and
[ShVe, (6.5)]). For i ∈ I 2 , the element

c ϑi = (c θi1 ⊠ Ei2 [c]) + (N c(0) ⊠ c θi2 ),

where ⊠ denotes the exterior product on motivic cohomology, has boundary ec . However, its
second term is problematic, as it is a unit on {0} × (Ei2 − Ei2 [c]), which contains (0, Pi2 ). We
can avoid this issue by a minor adjustment. We instead consider

µ∗i c ϑi = (c θi1 ⊠ Ei2 [c]) + µ∗i (N c(0) ⊠ c θi2 ), (3.4)

for a choice of matrix µi = ( x1 10 ) with x ∈ ai2 ,i1 c prime to N , as the second term is then
supported on ({0} × (Ei2 − Ei2 [c]))µ−1 ∗
i , while µi leaves its residue unchanged.
We may then view c Θi as the class of the cocycle

c Θi : Γi → K2 (i)(0)

uniquely determined by
∂(c Θi (γ)) = (γ ∗ − 1)µ∗i (c ϑi )
for γ ∈ Γi . Whether c Θ is being used to denote a collection of cocycles or their classes should
be gleaned from context: for instance, when studying the action of Hecke operators, we are
referring to the class, whereas when speaking of the values of c Θi , we are referring to the
cocycle.
Moreover, we have the following.

Lemma 3.4.1. The restriction of c Θi to Γ0 (N )i takes values in K2,N (i).

Proof. For γ = ( ac db ) ∈ Γi , set Sγ = (Ei1 × Ei2 [c])γ −1 and Sγ′ = ({0} × Ei2 )γ −1 . Our
choice of cocycle c Θi is determined uniquely by the fact that its values are trace fixed and
the residue of its value on γ ∈ Γi is (γ ∗ − 1)µ∗i c ϑi . This value lies in H 2 (Vγ , Z(2))(0) for
Vγ the complement in Ei of the four codimension one subvarieties S1 , Sγ , Sµ′ i , and Sγµ′
i
. For
a b
γ = ( c d ) ∈ Γ0 (N )i , each of 1, d, x and c + dx is nonzero modulo N , so this cohomology
group is a subgroup of K2,N (i)(0) .

25
Remark 3.4.2. The group K2,N (i) fits in a ∆0 (N )i -equivariant subcomplex KN (i) of K(i)
which in degrees p ∈ {1, 0} consists of the trace-fixed part of the sums of K-groups of the
irreducible p-cycles not intersecting the (O/N )× -orbit of (0, Pi ) in Ei . Together for all i, these
give a ∆0 (N )-module system. Since c ei ∈ K0,N (i) and µ∗i c ϑi ∈ K1,N (i), using the complex
KN (i), we get that the class c Θi |Γ0 (N )i and its explicit representative have canonical lifts valued
in K2,N (i). In particular, since the double coset decompositions of Tp are unchanged upon
passage from the groups Γi to their subgroups Γ0 (N )i for primes p ∤ N , we still have the
Eisenstein property of c Θi |Γ0 (N )i viewed as a class in H 1 (Γ0 (N )i , K2,N (i)) for such operators.
The above construction of c Θi can be carried out for any product A = A1 × A2 of elliptic
curves with CM by O, defined over some extension of the Hilbert class field of F . As such,
the following lemma holds.

Lemma 3.4.3. Let A1 and A2 be elliptic curves with CM by O defined over a field L containing
∼
F (f), and suppose that there exist i ∈ I 2 and L-isomorphisms ψu : Au − → Eiu for u ∈ {1, 2}.
Let ψ = (ψ1 , ψ2 ), which induces an action of Γi on A = A1 × A2 . Then the cocycle
(0)
c ΘA : Γi → K2 (L(A)) attached to N c2 (0) − A[c], with residue the analogue of (3.4) for the
same µi , satisfies c ΘA = ψ ∗ c Θi .

4 Specialized cocycles for ray class fields

4.1 Pullback by N -torsion
Let us now choose N so that O× → (O/N )× is injective. For α ∈ F × prime to N and i ∈ I 2 ,
let [α]i : Ei [N ] → Ei [N ] be multiplication by any element in O congruent to α modulo N . Let
us view (0, Pi2 ) as a morphism λi : Spec F (N ∩ f) → Ei [N ]. It then defines a pullback map

λ∗i : K2,N (i) → K2 (F (N ∩ f))Z′ .

Lemma 4.1.1. The image of λ∗i ◦ c Θi |Γ0 (N )i is Gal(F (N ∩ f)/F (N ))-fixed.

Proof. For j ∈ {1, 2}, choose isomorphisms ψj : Aj → Eij with elliptic curves Aj with CM
by O that are defined over F (N ) and have torsion contained in F ab . These are necessarily
F (N ∩ f)-isomorphisms (noting Lemma 3.1.1) as the compositions of the Hecke characters
of Aj and Eij with norms from F (N ∩ f) agree by part (i) of the lemma of [deS, 1.4]. Note in
particular that the N -torsion in each Aj is defined over F (N ). If P = ψ2−1 (Pi2 ), then by and
in the notation of Lemma 3.4.3, we have

(0, P )∗ ◦ c ΘA = (0, P )∗ ◦ ψ ∗ ◦ c Θi = λ∗i ◦ c Θi

26
in the sense that λ∗i ◦ c Θi : Γ0 (N )i → K2 (F (N ∩f))Z′ takes values in the image of the codomain
of (0, P )∗ ◦ c ΘA , which is K2 (F (N ))Z′ .

We now take f to be relatively prime to N .

Proposition 4.1.2. For i, j ∈ I 2 and ( ac db ) ∈ ∆0 (N )i,j , we have the equality

∗
λ∗i ◦ ( ac db ) = R(d′ ) ◦ λ∗j

of specialization maps on K2,N (j), where d′ ∈ O is such that d′ ≡ d mod N and d′ ≡ 1 mod f.
∗
Proof. First, note that λ∗i ◦ ( ac db ) = (( ac db ) ◦ λi )∗ agrees with pullback by (0, d · Pi2 ). Corollary
3.1.3 and Lemma 3.1.4 tell us that

d · Pi2 = [d]j2 Pj2 = [d′ ]j2 Pj2 = R(d′ )(Pj2 ).

We therefore have the first equality in

∗
λ∗i ◦ ( ac db ) = (0, R(d′ )(Pj2 ))∗ = R(d′ ) ◦ λ∗j ,

where for the second equality, we have used the fact that K2,N (i) is generated by classes of
cycles defined over F (f), which are fixed under the action of R(d′ ).

Let RN denote the Artin map from the N -ray class group ClN (F ) of F to Gal(F (N )/F ).
Lemma 4.1.1 and Proposition 4.1.2 combine to provide the following.

Corollary 4.1.3. For i, j ∈ I 2 and ( ac db ) ∈ ∆0 (N )i,j , we have

∗
λ∗i ◦ ( ac db ) ◦ c Θj = RN (d) ◦ λ∗j ◦ c Θj

on Γ0 (N )j .

We then have a specialized cocycle that is independent of f.

Proposition 4.1.4. There exists a cocycle

c Θi,N : Γ0 (N )i → K2 (F (N ))Z′ .

such that for every choice of f, the cocycles c Θi,N and λ∗i ◦ c Θi agree as maps to K2 (F (N f)) ⊗
Z′ [ f1 ], where f is the order of (O/f)× .

27
Proof. Since [F (N f) : F (N )] divides f , we have

K2 (F (N f))Gal(F (N f)/F (N )) ⊗ Z′ [ f1 ] ∼
= K2 (F (N )) ⊗ Z′ [ f1 ].

By Lemma 4.1.1, we may therefore speak of the pullback λ∗i ◦ c Θi as taking values in
K2 (F (N )) ⊗ Z′ [ f1 ] for i ∈ I 2 . It is then a cocycle for the action of ( ac db ) ∈ Γi on K2 (F (N ))
by RN (d) by Corollary 4.1.3. By construction and Lemma 3.4.3, this cocycle is, up to the
inversion of f , independent of the choice of f. Varying f (or even just taking it to be a suffi-
ciently large power of a prime over 2 prime to N if such a prime exists), we obtain the claimed
well-defined cocycle.

The following corollary is then immediate from the definitions.

Corollary 4.1.5. For i, j ∈ I 2 and g ∈ ∆0 (N )i,j , we have

T (g)(c Θj,N ) = λ∗i ◦ T (g)(c Θj |Γ0 (N )j )

in H 1 (Γ0 (N )i , K2 (F (N ))Z′ ).

Proposition 3.3.3, noting Remark 3.4.2, translates to give the Eisenstein property of the
specialized cocycles. For this, note that if p is prime to N and ai1 ,j1 p = (η) and i2 = j2 , then



the double cosets Γi η0 10 Γj and Γ0 (N )i η0 10 Γ0 (N )j have sets of left coset representatives
that are equal.

Corollary 4.1.6. For any prime ideal p of O not dividing N , the operator Tp − (N p + [p]∗ )
annihilates c ΘN = (c Θi,N )i∈I 2 , and every diamond operator ⟨d⟩∗ acts trivially on c ΘN .

The following lemma will be useful for us.

Lemma 4.1.7. Let i, j ∈ I 2 and an ideal n of O prime to N c be such that i ∼n j. For

u ∈ {1, 2}, let ηj ∈ F × be a generator of aiu ,ju n. Then the identity

λ∗j ◦ η01 η02 ∗ ◦ c Θi = RN (ai2 ,j2 )−1 ◦ λ∗i ◦ c Θi

is satisfied on Γ0 (N )i .

Proof. For u ∈ {1, 2}, the theory of complex multiplication as in [deS, 1.5] provides unique
R(n)
F (f)-isogenies Eiu → Eiu with kernel Eiu [n] that agree with R(n) on prime-to-n torsion. Let
(n) R(n) R(n)
ϕi : Ei → Ei denote the resulting isogeny. We then have an isomorphism ψ : Ei → Ej
(n) (n) (n)
such that ψ ◦ ϕi = ρ : Ei → Ej . Setting λi = ϕi ◦ λi , Corollary 3.1.3 and Lemma 3.1.4
(n)
imply that ψ ◦ λi = R(η2′ ) ◦ λj for any η2′ ∈ O with η2′ ≡ 1 mod f and η2′ ≡ η2 mod N . We
then have
(n) (n) (n)
(R(η2′ )λj )∗ ◦ ρ∗ = (R(η2′ )λj )∗ ◦ ψ∗ ◦ (ϕi )∗ = (λi )∗ ◦ (ϕi )∗ (4.1)

28
on K2,N (i).
R(n)
We have a canonical identification of Aut(Ei ) with Γi given by applying the Galois
element R(n) to the automorphism of Ei defined by an element of Γi . This gives the motivic
R(n) (n)
complex K′ (i) = K(Ei ) a left pullback action of Γi . Pushforward by the isogeny ϕi
induces a morphism of complexes of Γi -modules between K(i) and K′ (i), as does application
R(n)
of the Galois element R(n) by the choice of the Γi -action on Ei we have taken. We let
′ (n)
c Θi = (ϕi )∗c Θi : Γi → K′2 (i),

which is the the unique cocycle satisfying

(n)
∂(c Θ′i (γ)) = (γ ∗ − 1)(ϕi )∗ µ∗i (c ϑi ).

We claim that we have the following equality of cocycles on Γi :

′
c Θi = R(n) ◦ c Θi . (4.2)
(n)
This reduces quickly to the equality (ϕi )∗c ϑi = R(n)c ϑi in K′1 (i). Note that c ϑi is a sum
of elements of H 1 (Ei1 [c] × (Ei2 − Ei2 [c]), 1)(0) and H 1 ((Ei1 − Ei1 [c]) × Ei2 [c], 1)(0) , and
the residue map from each of these groups to H 0 (Ei [c], 0)(0) is injective. Thus, we have the
(n)
equality from the agreement of ϕi and R(n) on prime-to-n torsion.
(n)
Restricting the equality (4.2) to Γ0 (N )i and noting that R(n)(λi ) = λi , we see noting
Lemma 4.1.1 that
(n)
(λi )∗ ◦ c Θ′i = RN (n) ◦ λ∗i ◦ c Θi . (4.3)
As RN (η2 ) = RN (η2′ ) and RN (n)RN (η2 )−1 = RN (ai2 ,j2 )−1 , combining (4.3) with (4.1)
yields the desired identity.

The following provides a direct connection between the classes c Θi,N for equivalent i ∈ I 2 .

Proposition 4.1.8. For i, j ∈ I 2 and n an ideal of O prime to N such that i ∼n j, we have

[n]∗ (c Θj,N ) = RN (n) ◦ c Θi,N

in H 1 (Γ0 (N )i , K2 (F (N ))Z′ ).

Proof. We verify this in H 1 (Γ0 (N )i , K2 (F (N ))) ⊗Z Z′ [ f1 ] for f = |(O/f)× | by working with

the tuple (λ∗k ◦ c Θk )k∈I 2 for a given f prime to N , as it agrees with (c Θk,N )k∈I 2 upon inverting
f . The result then follows by varying f.
Choose an integral ideal n′ coprime to N c having the same ideal class as n. Write
aiu ,ju n = (ηu ) and aiu ,ju n′ = (ηu′ ) for u ∈ {1, 2}, chosen such that η2′ η1 = η2 η1′ . Set

29
  
η1 0 η1′ 0
and ρ′ = . Recall from Lemma 3.3.1 that [n′ ]∗ (c Θi ) = c Θj as cohomology


ρ= 0 η2 0 η2′
classes. Since ρ′ ρ−1 is scalar, we also have

[n]∗ ([n′ ]∗ (c Θi ))(γ) = ρ∗ ◦ ρ′∗ ◦ c Θi (ρ′ ρ−1 γρ(ρ′ )−1 )

= ρ∗ ◦ ρ′∗ ◦ c Θi (γ)

for γ ∈ Γi . Combining these identities with Corollary 4.1.5, Proposition 4.1.2 and Lemma
4.1.7, we have

[n]∗ (c Θj,N ) = λ∗i ◦ [n]∗ (c Θj )

= λ∗i ◦ [n]∗ ([n′ ]∗ (c Θi ))
= λ∗i ◦ ρ∗ ◦ ρ′∗ ◦ c Θi
= R(nai2 ,j2 ) ◦ λ∗j ◦ ρ′∗ ◦ c Θi
= RN (n) ◦ c Θi,N .

This allows us to describe the relationship between the classes c Θi,N as we vary c.

Corollary 4.1.9. Let c and d be ideals of O prime to N . Then we have the equality of classes

(N d2 − RN (d))c Θi,N = (N c2 − RN (c))d Θi,N

for each i ∈ I 2 .

Proof. Proposition 4.1.8 and Lemma 3.3.6 combine to tell us that

RN (d) ◦ c Θi,N = cd Θi,N − N c2 d Θi,N ,

and similarly with c and d reversed, from which the identity follows.

4.2 Integrality
Fix an ideal c of O prime to N f. Adopting the notation of the proof of Lemma 3.4.1, we set
T = S1 ∪ Sµ′ i . For γ ∈ Γi , we set Tγ = T ∪ T γ −1 and Vγ = Ei − Tγ , and we let Tγ◦ be the
complement in Tγ of the pairwise intersections of S1 , Sγ , Sµ′ i , and Sγµ
′
i
.
The elliptic curves Er for r ∈ I have good reduction at all primes of the ring of integers
O(f) of F (f) not dividing f, as f is the conductor of the Hecke character of Er . Let ei denote
the reduction of Ei modulo a prime l of O(f) not dividing f. Let R be the localization of O(f)
at l, and let Ei/R be the product of the Neron models of Ei1 and Ei2 over R. We use vγ and

30
t◦γ (resp., Vγ/R and Tγ/R
◦
) in place of Vγ and Tγ◦ , respectively, to denote the subschemes of ei
(resp., Ei/R ) constructed in the same manner as the so-denoted subschemes of Ei .
We continue to assume that f is prime to N .
Lemma 4.2.1. We have a commutative square of residue maps

H 2 (Vγ , 2) H 1 (vγ , 1)
(4.4)

H 1 (Tγ◦ , 1) H 0 (t◦γ , 0)

that commute with trace maps [α]∗ for α ∈ O prime to l.

Proof. By [Lev, Theorem 1.7], We have a distinguished triangle of cycle complexes computing
motivic cohomology:

0 → zq (vγ , ∗) → zq (Vγ/R , ∗) → zq (Vγ , ∗) (4.5)

and similarly for Tγ◦ . This gives the horizontal residue maps in (4.4). Since the maps in (4.5)
commute with those induced by multiplication by α ∈ O prime to l, we have the commutativity
of the horizontal maps with [α]∗ . The vertical maps in (4.4) are the more usual residue maps
over a common base field, which we already know commute with these trace maps.
Lemma 4.2.2. Let l be a prime of O(f) not dividing f. For each γ ∈ Γi , the value c Θi (γ) lies
in the kernel of the residue map H 2 (Vγ , Z′ (2)) → H 1 (vγ , Z′ (1)).
Proof. In the diagram (4.4), the two vertical maps are injective upon taking trace-fixed parts.
To see this for the right-hand vertical map, the relevant coniveau spectral sequence (as employed
in [ShVe, Section 2.2]) tells us that the kernel is H 1 (ei , 1)(0) , where ei is the reduction of Ei
modulo l. Since ei is projective, this group consists only of constant units, but as these are
pulled back from the residue field, [α]∗ acts on them by raising to the power N α. Thus, only
the identity in H 1 (ei , 1)(0) is trace-fixed.
Recall that c Θi (γ) has residue (γ ∗ − 1)µ∗i c ϑi ∈ H 1 (Tγ◦ , 1)(0) for γ ∈ Γi . By Lemma 4.2.1
and the injectivity of the righthand vertical map on trace-fixed parts, we need only show that
each γ ∗ (c ϑi ) for γ ∈ Γi has trivial residue in H 0 (t◦γ , 0). By the relevant Gysin sequence as in
[Gei, Corollary 3.4], it then suffices to check that each γ ∗ (c ϑi ) for γ ∈ Γi is the restriction of a
◦
unit on Tγ/R .
◦
It is enough to see that c ϑi is a unit on T/R , where T ◦ = T1◦ . Since l does not divide the
conductor f of the Hecke characters of Ei1 and Ei2 , these elliptic curves have good reduction
at l. Then c ϑi extends over relevant open subscheme of the fiber product of Neron models of
Ei1 and Ei2 over the localization of O(f) at l, since the theta functions used in its definition
(3.4) do, as desired.

31
 For any multiple M of N , let us write

O(M)[ 1 ] if N is a prime power,
′ N
O (M) =
O(M) otherwise

for brevity of notation in what follows.

Lemma 4.2.3. Let l be a prime of O′ (N f) not lying over f. For each γ ∈ Γ0 (N )i and primitive
N -torsion point P of Ei2 , the F (N f)-point (0, P ) ∈ Vγ reduces to a point of vγ .

Proof. It suffices to show that the mod l reduction of (0, P ) does not lie in the reduction of
T ρ−1 for ρ ∈ Γ0 (N )i , as Tρ = T ∪ T ρ−1 . We can consider ρ = 1 by replacing (0, P ) by
(0, Pi2 )ρ−1 = (0, P ′ ) for some primitive N -torsion point P ′ ∈ Ei2 . It further suffices to show
that the reduction of (0, P ) does not lie in the reduction of S1 = Ei1 ×Ei2 [c], and, translating by
µ−1 1 0 ′
c = ( −x 1 ), that the reduction of (−xP, P ) does not lie in the reduction of S1 = {0} × Ei2 .
The former amounts to showing that no point of P + Ei2 [c] reduces to zero at l, which follows
from [Ru, Lemma 7.3(ii)]. For the latter, we need only note that x is prime to N and P is
nonzero modulo l by the same lemma.

Proposition 4.2.4. The cocycle c Θi,N takes values in K2 (O′ (N ))Z′ .

Proof. Let l be a prime of O′ (N f) not dividing f. By Lemma 4.2.3, the reduction of (0, Pi2 )
lies in vγ for each γ ∈ Γ0 (N )i . We apply Lemma 4.2.2, pull back by this reduced point and
use the commutativity of pullbacks and residues to see that c Θi,N (γ) has trivial residue in the
tensor product with Z′ of the first K-group of the residue field of O′ (N f) at l.
We thus get that c Θi,N (γ) lies in K2 (O′ (N f)[ 1f ])Z′ , but we also know it is fixed by
Gal(F (N f)/F (N )), so again by Galois descent it takes values in K2 (O′ (N )) ⊗ Z′ [ f1 ] for
f = |(O/f)× |. Since the value c Θi,N (γ) is independent of the choice of f in the sense of
Proposition 4.1.4, we therefore have that it lies in K2 (O′ (N ))Z′ .

In fact, we claim that the class c Θi,N is Eisenstein with values in this smaller group. The
key point is the following lemma.

Lemma 4.2.5. For any nonzero ideal d of O, the map

H 1 (Γ0 (N )i , K2 (O(N )[ 1d ])) → H 1 (Γ0 (N )i , K2 (F (N )))

is injective.

32
Proof. Recall that H denotes the Hilbert class field of F . Consider the commutative square

kq×
L
K2 (H) q∤d

K2 (F (N ))Gal(F (N )/H) ( Q∤d k(N )× Gal(F (N )/H)

L
Q) ,

where kq (resp., k(N )Q ) denotes the residue field of a prime q of H (resp., Q of F (N )). The
upper horizontal arrow is the surjection in the standard localization sequence in K-theory, and
the right-hand vertical arrow is clearly an isomorphism. Thus, the lower horizontal map is
surjective as well, which yields the desired injectivity, as Γ0 (N )i acts on the groups in question
through its surjective image in Gal(F (N )/H).

This yields the Eisenstein property of the integral cocycles c Θi,N as a corollary.

Corollary 4.2.6. The collection c ΘN ∈ i∈I 2 H 1 (Γ0 (N )i , K2 (O′ (N ))Z′ ) is Eisenstein away
L

from N . That is, it is annihilated by Tp − (N p + RN (p)) for all primes p of O not dividing N .

Proof. The stated Eisenstein property, but for cohomology with coefficients in K2 (F (N ))Z′ ,
is Corollary 4.1.6 and Proposition 4.1.8. We then apply Lemma 4.2.5.

4.3 Unaugmented cocycles

Let p ≥ 7 be a prime number. Write Q = Gal(F (N )/F ) as a product Qp × Q′ of its Sylow
p-subgroup Qp and its maximal prime-to-p order subgroup Q′ . The maximal ideals of Zp [Q]
correspond to GQp -conjugacy classes of p-adic characters of Q′ .
Fix a p-adic character χ of Q′ , and let Oχ be the Zp -algebra generated by its image. The
localization of Zp [Q] determined by χ is isomorphic to Oχ [Qp ], with the projection map
χ̃ : Zp [Q] → Oχ [Qp ] coming from the Zp [Qp ]-linear extension of χ. For a Zp [Q]-module M ,
let us define the χ-component of M as

M (χ) = M ⊗Zp [Q] Oχ [Qp ]

where the right tensor product is given by χ̃. Though defined as a quotient, M (χ) is also a
direct summand of M via the idempotent determined by χ̃.
In the following, we extend χ to Q by taking it to be trivial on Qp . We then view it as a
character of ClN (F ) via the Artin map RN .

Lemma 4.3.1. Let c be an ideal of O prime to N . The projection of N c2 − RN (c) to Zp [Q](χ)

is a unit if and only if χ(c) ̸≡ N c2 mod pOχ .

33
Proof. The element N c2 − RN (c) projects to a unit in the χ-component of Zp [Q] if and only
if it reduces to a unit in the coinvariant group for Qp , which is isomorphic to Oχ . Equivalently,
N c2 − RN (c) is a unit if and only if N c2 − χ(c) is. As Oχ is an unramified extension of Zp ,
we have the first statement.

If all primes over p divide N , which is to say (p) | N 2 , then we let ω : Q′ → Z× p be the
composition of restriction to Q(µp ) with the canonical injection Gal(Q(µp )/Q) ,→ Z× p that is
the unique lift of the modulo p cyclotomic character. The character induced by ω on ClN (F )
agrees modulo p with the reduction of the norm map.
In what follows, we shall take χ ̸= ω 2 as being automatically satisfied if (p) ∤ N 2 .

Corollary 4.3.2. If χ ̸= ω 2 , then there exists an ideal c of O prime to N such that the projection
of N c2 − RN (c) to Zp [Q](χ) is a unit.

Let
χ
c Θi,N : Γ0 (N )i → (K2 (O′ (N )) ⊗ Zp )(χ)
be the cocycle given by composing c Θi,N with projection to the χ-component of the p-part of
K2 (O′ (N )).

Theorem 4.3.3. For all χ ̸= ω 2 and each i ∈ I 2 , there is a unique class

Θχi,N ∈ H 1 (Γ0 (N )i , (K2 (O′ (N )) ⊗ Zp )(χ) ).

such that
(N c2 − RN (c))Θχi,N = c Θχi,N (4.6)
for every ideal c of O prime to N . Moreover, the collection ΘχN = (Θχi,N )i∈I 2 is Eisenstein
away from N .

Proof. We define
Θχi,N = (N d2 − RN (d))−1 d Θχi,N
in H 1 (Γ0 (N )i , (K2 (O′ (N )) ⊗ Zp )(χ) ) for d such that N d2 − χ(d) is a p-adic unit, which
exists by Corollary 4.3.2. The property (4.6) of Θχi,N follows from Corollary 4.1.9 and Lemma
4.2.5 and clearly implies uniqueness. The tuple ΘχN of classes is Eisenstein away from N by
Corollary 4.2.6.
2
Remark 4.3.4. Even for χ = ω 2 , we can make sense of p(N c2 − RN (c))−1 c Θωi,N for a good
2
choice of c. That is, what we might denote pΘωi,N is well-defined, even if it is not clear that
2
Θωi,N is.

34
 If we can construct such Θχi,N for all χ (including ω 2 if (p) | N 2 ), then we can sum them
to obtain an unaugmented class Θi,N on p-parts, as in the following theorem. Recall that h is
the class number of F .

Theorem 4.3.5. Suppose that either (p) ∤ N 2 or p ∤ h. For i ∈ I 2 , there exists a unique class
Θi,N ∈ H 1 (Γ0 (N )i , K2 (O′ (N )) ⊗ Zp ) such that

(N c2 − RN (c))Θi,N = c Θi,N

for all ideals c of O prime to N . The collection ΘN = (Θi,N )i∈I 2 is Eisenstein away from N .

Proof. If (p) ∤ N 2 , then the result follows from Theorem 4.3.3, since the condition χ ̸= ω 2 is
automatically satisfied. We therefore suppose that (p) | N 2 and p ∤ h, in which case it similarly
suffices to show that the ω 2 -component of K2 (O′ (N )) ⊗ Zp is trivial.
Note that either N is not a prime power or p is non-split in F and N is a power of the
prime over p. In the former case, O′ (N ) = O(N ), and in the latter case K2 (O′ (N )) ⊗ Zp ∼ =
K2 (O(N )) ⊗ Zp . Now, by a classical result of Tate [Tat, Theorem 5.4], we have

K2 (O(N )) ⊗ Zp ∼ 2
= Hét (O(N )[ p1 ], Zp (2)).

Let us show that the ω 2 -component of the latter cohomology group is trivial if p ∤ h by
showing that its quotient by the action of the maximal ideal of Zp [Qp ] is. Since the Galois
group of the maximal unramified outside p-extension of a number field has p-cohomological
dimension 2 (cf. [NSW, Lemma 3.3.11]), corestriction gives the first isomorphism (cf. [NSW,
Theorem 10.2.3]) in

(O(N )[ p1 ], Zp (2))(ω ) ⊗Zp [Qp ] Fp ∼ (ω 2 ) ∼

2
2
Hét 2
= Hét (O[µp , p1 ], µ⊗2
p )
2
= Hét (O[µp , p1 ], µp )(ω) .

Recall that Kummer theory provides an exact sequence

M
0 → Cl′ (F (µp )) ⊗ Fp → Hét
2
(O[µp , p1 ], µp ) → Fp → Fp → 0
v|p

(cf. [NSW, Proposition 8.3.11]), where here Cl′ (F (µp )) denotes the quotient of Cl(F (µp ))
by the classes of primes over p. As a Zp [Gal(F (µp )/Q)]-module, the ω-component of
this cohomology group breaks up as a sum of two components, that for the composition
ωQ : Gal(F (µp )/Q) → Z× p of the mod p cyclotomic character with the lift of the reduction
mod p map, and that for the product of ωQ and the p-adic character χF of the imaginary quadratic
field F . It follows from a quick examination of these groups that Hét2
(O[µp , p1 ], µp )(ω) is just
the ωQ χF -component of Cl′ (F (µp )) ⊗ Fp for the action of Gal(F (µp )/Q). By Leopoldt’s
Spiegelungssatz, this component is trivial as Cl(F ) ⊗ Fp is.

35
 Quite frequently, then, we have that the conditions of the following corollary are satisfied.
In fact, the above proof shows that the weaker condition of the triviality of the ωQ χF -component
of Cl′ (F (µp )) ⊗ Fp can be used to replace the condition of p dividing h.

Corollary 4.3.6. Suppose that there are no primes greater than 5 dividing both N 2 and h. For
i ∈ I 2 , there exists a unique class Θi,N ∈ H 1 (Γ0 (N )i , K2 (O′ (N ))Z′ ) such that

(N c2 − RN (c))Θi,N = c Θi,N

for all ideals c of O prime to N . The collection ΘN = (Θi,N )i∈I 2 is Eisenstein away from N .

5 Eisenstein maps on homology

5.1 Cohomology of Bianchi spaces
Let us quickly review the discussion of Section 2.2 in our setting of interest. Let H =
H2,F = C × R>0 denote the complex upper half-space, with the usual action of GL2 (F ).
We view I as the subset of I 2 consisting of pairs (r, 1) for r ∈ I. For r ∈ I, we then have
Γ1 (N )r = Γ1 (N )(r,1) and similarly for other subscripts. The Bianchi space for F of level
`
Γ1 (N ) is the disjoint union Y1 (N ) = r∈I Y1 (N )r , where Y1 (N )r = Γ1 (N )r \H. Then
Y1 (N ) also has the usual adelic description

Y1 (N ) = GL2 (F )\(GL2 (AfF ) × H)/U1 (N ).

Any element of finite order in Γ1 (N )r has order dividing 120, in that its eigenvalues are
roots of unity with sum and product in F , which are therefore contained in a number field of
degree 4. As in (2.3), for any ∆0 (N )-module system A = (Ar )r∈I indexed by I such that 5!
acts invertibly and scalar elements act trivially on each Ar , we have a canonical isomorphism

H 1 (Y1 (N ), A) ∼
M
= H 1 (Γ1 (N )r , Ar ). (5.1)
r∈I

This isomorphism is Hecke equivariant for the operators defined in and following Definition
2.3.4 by Proposition 2.2.2. On the right, if ar,s n is principal, then we have Tn = Tr,s (n, 1)
by Lemma 2.3.8(b). If ar,s n2 is principal, then we set Sn = Tr,s (n, n). Let us write ⟨n⟩ for
(⟨n⟩∗ )−1 . By Lemma 2.3.8(a), we have Sn = ⟨n⟩[n]∗ = [n]∗ ⟨n⟩.
If the action of Γ1 (N )r on Ar is trivial, then we have a canonical isomorphism
∼
→ Hom(H1 (Γ1 (N )r , Z′ ), Ar ).
ϕr : H 1 (Γ1 (N )r , Ar ) −

36
Suppose that the Ar are all equal to a fixed A so that the maps ϕr assemble into an isomorphism
∼
→ Hom(H1 (Y1 (N ), Z′ ), A).
ϕ : H 1 (Y1 (N ), A) −

Suppose also that each g = ( ac db ) ∈ ∆0 (N )r,s provides a map g : A → A depending only on

the image of d in (O/N )× , independent of r, s ∈ I.
Given g ∈ ∆0 (N )r,s , and writing Γ1 (N )r gΓ1 (N )s = vt=1 gt Γ1 (N )s , we define T (g) on
`

x ∈ H1 (Y1 (N )r , Z′ ) by
v
X
T (g)x = gt† x ∈ H1 (Y1 (N )s , Z′ ),
t=1

where gt† denotes the adjoint matrix to gt . We then have the following identity, as the reader
may verify by a similar argument to that given in the proof of [ShVe, Theorem 4.3.7].

Lemma 5.1.1. For ξ ∈ H 1 (Y1 (N )s , A) and x ∈ H1 (Y1 (N )r , Z′ ), we have

ϕ(T (g)ξ)(x) = g · ϕ(ξ)(T (g)x).

When g is such that T (g) : H 1 (Y1 (N )s , A) → H 1 (Y1 (N )r , A) is the restriction of Tn , Sn ,

or ⟨n⟩ for some ideal n of O prime to N , we write Tn , Sn , or ⟨n⟩, respectively, for the operator
H1 (Y1 (N )r , Z′ ) → H1 (Y1 (N )s , Z′ ) given by T (g) and also for the corresponding operator on
H1 (Y1 (N ), Z′ ).
Let us consider the restriction of c Θχr,N to Γ1 (N )r . Note that the above conditions on
A = K2 (O′ (N ))Z′ are satisfied, and g = ( ac db ) ∈ ∆0 (N )r,s acts as σd on K2 (O′ (N )). The
restriction in question yields a homomorphism

c Πr,N : H1 (Y1 (N )r , Z′ ) → K2 (O′ (N ))Z′ .

We can then take the sum

c ΠN : H1 (Y1 (N ), Z′ ) → K2 (O′ (N ))Z′

of these maps over r ∈ I.

For a prime p ≥ 7 and character χ ̸= ω 2 as in Section 4.3, we also have maps

Πχr,N : H1 (Y1 (N )r , Zp ) → (K2 (O′ (N )) ⊗ Zp )(χ) (5.2)

and ΠχN , and the analogues of the statements which follow also hold for these.

37
Proposition 5.1.2. For r, s ∈ I and a nonzero ideal n of O prime to N such that ar,s n2 is
principal, we have
c Πs,N ◦ Sn = RN (n) ◦ c Πr,N .

In particular, for d ∈ (O/N )× , we have

c Πr,N ◦ ⟨d⟩ = RN (d) ◦ c Πr,N .

Proof. Let q ∈ I be such that ar,q n and as,q n−1 are principal. Let u ∈ I be such that au n is
principal. Lemma 2.3.8(a), Corollary 3.3.5, and Lemma 4.1.8 tell us that

Sn (c Θs,N ) = [n]∗ ⟨n⟩(c Θs,N ) = [n]∗ (c Θ(q,u),N ) = RN (n) ◦ Θr,N .

On the other hand, note that the operator Sn = [n]∗ ⟨n⟩ on cohomology is given by pullback
by a matrix in ∆0 (N )r,s with lower right-hand entry congruent to a unit modulo N . We then
need only apply Lemma 5.1.1 for ξ corresponding to c Θs,N and T (g) = Sn to obtain the first
statement. The last statement follows by observing that [d]∗ , the pullback by a scalar matrix,
acts trivially on H1 (Y1 (N )r , Z′ ).

The Eisenstein property of the maps c ΠN is a consequence of Corollary 4.1.6: that is,

c ΠN ◦ (Tp − N p − Sp ) = 0 (5.3)

for every prime ideal p of O not dividing N . Together with Proposition 5.1.2, we may rephrase
this as the following.

Proposition 5.1.3. For every prime ideal p of O not dividing N , we have

c ΠN ◦ Tp = (N p + RN (p)) ◦ c ΠN .

More precisely,
c Πs,N ◦ Tp = (N p + RN (p)) ◦ c Πr,N
for r, s ∈ I such that ar,s p is principal.

We expect that the maps on the homology of Y1 (N ) factor through the homology of the
compactification X1 (N ) given by adjoining cusps. We prove this for certain χ in the next
subsection.

38
5.2 Borel-Serre boundary
The Borel-Serre compactification X1BS (N ) of Y1 (N ) can be written as

X1BS (N ) = GL2 (F )\(GL2 (AfF ) × H)/U1 (N )

where H = H ∪ ∂H with ∂H a disjoint union of components indexed by α ∈ P1 (F ) that can be

thought of as placing (0, 1] × (P1 (C) − {α}) at α. Here, an element of GL2 (F ) acts on (α, t, x)
for t ∈ (0, 1) and x ∈ P1 (C) as a Möbius transformation on the first and last coordinates.
The rth component is X1BS (N )r = Γ1 (N )r \H. The embedding of Y1 (N ) in X1BS (N ) is a
homotopy equivalence.
For B the upper-triangular Borel subgroup of G = GL2 (F ), we have G/B ∼ = P1 (F )
via the G-equivariant map sending the identity to ∞. Given x ∈ P1 (F ) and any τx ∈ G
mapping to x, the group Bx,r = Γ1 (N )r ∩ τx Bτx−1 is the stabilizer of x in Γ1 (N )r . Letting
C1 (N )r = Γ1 (N )r \P1 (F ), we also use Bx,r for x ∈ C1 (N )r to denote Bx̃,r for a choice of
x̃ ∈ P1 (F ) lifting the cusp x. As in the discussion of [Har1, p. 48–49] (see also [Ber, p. 20]),
the space a
Bx,r \H (5.4)
x∈C1 (N )r

is homotopy equivalent to ∂X1BS (N )r . This yields the following description of the singular
first cohomology of the Borel-Serre boundary:

H 1 (∂X1BS (N ), Z′ ) ∼
M M
= H 1 (Bx,r , Z′ ). (5.5)
r∈I x∈C1 (N )r

In particular, H 1 (∂X1BS (N ), Z′ ) is torsion-free.

For a ∆-module system (Ar )r∈I , we can define the sheaf A on X1BS (N ) as we did on
Y1 (N ), and this agrees with the pushforward sheaf from the latter space. We then have Hecke
actions on the cohomology of X1BS (N ) and ∂X1BS (N ) with A-coefficients, and the maps on
cohomology induced by pullback of the embedding of Y1 (N ) into X1BS (N ) are equivariant for
the actions of Hecke operators.
The first interior cohomology group H!1 (Y1 (N ), A) of Y1 (N ) with A-coefficients is the
image of the second map, or kernel of the third, in the exact sequence

H 0 (∂X1BS (N ), A) → Hc1 (Y1 (N ), A) → H 1 (Y1 (N ), A) → H 1 (∂X1BS (N ), A). (5.6)

By (5.5) and the fact that Γ1 (N )r has no elements of prime order at least 7, this interior
cohomology group can be identified with the sum over r ∈ I of the parabolic cohomology
groups  M 
HP1 (Γ1 (N )r , A) = ker H 1 (Γ1 (N )r , A) → H 1 (P, A) ,
P

39
where P runs over (representatives of Γ1 (N )r -conjugacy classes of) parabolic subgroups of
Γ1 (N )r , or equivalently, stabilizers Bx,r of chosen lifts of cusps x ∈ C1 (N )r on the rth
component X1 (N )r of the Satake compactification X1 (N ) of Y1 (N ). In fact, we have the
following.
Lemma 5.2.1. We have canonical isomorphisms

= H!1 (Y1 (N ), A) ∼
Hom(H1 (X1 (N ), Z′ ), A) ∼
M
= HP1 (Γ1 (N )r , A),
r∈I

compatible with Hecke actions.

`
Proof. Let C1 (N ) = r∈I C1 (N )r denote the zero-dimensional space of cusps on X1 (N ).
Poincaré duality and the universal coefficient theorem (again using that 30 is invertible in Z′ )
yield canonical isomorphisms

Hom(H1 (X1BS (N ), ∂X1BS (N ), Z′ ), A) ∼

= Hom(H 2 (Y1 (N ), Z′ ), A) ∼
= Hc1 (Y1 (N ), A).
This allows us to identify the first map in the exact sequence (5.6) with the upper horizontal
map in the commutative square

Hom(H0 (∂X1BS (N ), Z′ ), A) Hom(H1 (X1BS (N ), ∂X1BS (N ), Z′ ), A)

≀ ≀

Hom(H0 (C1 (N ), Z′ ), A) Hom(H1 (X1 (N ), C1 (N ), Z′ ), A).

The cokernel of the lower horizontal map is Hom(H1 (X1 (N ), Z′ ), A), which gives the first
isomorphism, the second already having been explained. The maps in question are Hecke-
equivariant by construction and our earlier discussions.

Let IN be the prime-to-N ideal group of F . We use the convention that a Hecke character of
conductor m dividing N and infinity type (j, k) with j, k ∈ Z is a homomorphism ϕ : IN → C×
such that for x ∈ O with x ≡ 1 mod m, we have ϕ((x)) = xj x̄k , and m is the largest ideal with
this property. Such a ϕ gives rise to a map ϕ̃ : A× F /F
×
→ C× such that the restriction of ϕ̃ to
the finite ideles induces ϕ and which at the infinite place is just C× → C× by z 7→ z −j z̄ −k .
Let Z (resp. Z ′ ) denote the set of pairs µ = (µ1 , µ2 ) of Hecke characters IN → C× of
respective infinity types (−1, 0) and (1, 0) (resp. (0, −1) and (0, 1)) such that the product of the
conductors f1 and f2 of µ1 and µ2 divides N . We view µ ∈ Z∪Z ′ as a map B(F )\B(AF ) → C×
that sends the class of a matrix ( a0 db ) to µ1 (a)µ2 (d).
For a commutative ring C, let HC (N ) denote the Hecke C-algebra of endomorphisms of
H (Y1 (N ), C) ⊕ H 1 (∂X1BS (N ), C) generated by the operators Sp and Tp for primes p of O
1

not dividing N . For µ ∈ Z and an HC (N )-module M , let Mµ denote the maximal submodule
of A upon which each Sp acts as µ1 µ2 (p) and each Tp acts as µ1 (p)N p + µ2 (p).

40
Lemma 5.2.2. We have a direct sum decomposition

H 1 (∂X1BS (N ), C) ∼
M
= H 1 (∂X1BS (N ), C)µ
µ∈Z

of HC (N )-modules.

Proof. For µ ∈ Z ∪ Z ′ , let V (µ) denote the HC (N )-module of right U1 (N )-invariant maps
ψ : GL2 (AfF ) → C such that ψ(bg) = µ(b)ψ(g) for all b ∈ B(AfF ) and g ∈ G. By [Har2,
Theorem 1] (see also the discussion of [Ber, Section 2.10.1]), the isomorphism of (5.5) gives
rise to an isomorphism of HC (N )-modules

H 1 (∂X1BS (N ), C) ∼
M
= (V (µ) ⊕ V (µ′ )),
µ∈Z

where µ′ = (µ2 N −1 , µ1 N ) ∈ Z ′ . Here, N is the absolute norm, which is a Hecke character

of type (1, 1) having trivial conductor. Now, V (µ) decomposes as a restricted tensor product
V (µ) = ′p Vp (µ) over the primes p of O, and the action of Tp is trivial on all components
N

but Vp (µ). For p ∤ N , the latter representation is one-dimensional (being the right GL2 (Op )-
invariants of an unramified principal series). For the unique function ψp : GL2 (Fp ) → C
in Vp (µ) sending the identity to 1 (i.e., the spherical vector) and for π a uniformizer of the
valuation ring of Fp , we compute
X
(Tp ψp )(1) = ψp (( 10 π0 )) + ψp (( π0 1b )) = N pµ1 (p) + µ2 (p),
b̄∈O/p

so Tp acts as N pµ1 (p) + µ2 (p) on V (µ). Similarly, Sp acts as µ1 (p)µ2 (p). The complex
conjugate of µ′ = (µ′1 , µ′2 ) is in Z, and V (µ′ ) has the same Hecke action as V (µ).

Let p be an odd prime. Fix a prime P over p of the integer ring R of the number field
generated over F by the images of all µ1 and µ2 for (µ1 , µ2 ) ∈ Z. Let p = P ∩ O. The
restrictions to IN p of µ1 and µ2 take image in the units at P, so we may speak of their
reductions modulo P.
Let Zp denote the set of pairs ν = (ν1 , ν2 ) of Hecke characters ν1 , ν2 : IN p → F×
q for some
p-power q that are the respective reductions of µ1 and µ2 modulo P for some µ = (µ1 , µ2 ) ∈ Z.
Note that ν1 and ν2 factor through the quotient ClN ∩p (F ), and in fact through its prime-to-p
part. The conductor of νk for k ∈ {1, 2} divides fk ∩ p for all choices of µ ∈ Z reducing to ν.
Remark 5.2.3. For any prime ideal q of O, the group (O/q)× maps canonically to ClN ∩p∩q (F ),
which in turn has ClN ∩p (F ) as a canonical quotient. Thus, we can make sense of the restriction
of νk to (O/q)× .

41
 Let W denote the Witt vectors of Fp . We can view each νk for (ν1 , ν2 ) ∈ Zp as a character
valued in W × by taking its unique lift. With this convention, for an HW (N )-module A, let Aν
denote its localization at the maximal ideal containing Sq −ν1 ν2 (q) and Tq −(N qν1 (q)+ν2 (q))
for each prime q of O not dividing N p.

Proposition 5.2.4. There exists an isomorphism

H 1 (∂X1BS (N ), W ) ∼
M
= H 1 (∂X1BS (N ), W )ν ,
ν∈Zp

and similarly with W replaced with Fp .

Proof. Fix an embedding of Qp into C. For ν ∈ Zp , let Mν = H 1 (∂X1BS (N ), W )ν . This con-

tains the intersection with M = H 1 (∂X1BS (N ), W ) of the direct sum of the H 1 (∂X1BS (N ), C)µ
L
over µ ∈ Z reducing to ν. On the other hand, the sum ν∈Zp Mν of W -modules is a direct
summand of M , since the maximal ideals giving the localizations are distinct. This sum
contains M by Lemma 5.2.2, so we are done.

Let ξ : (O/p)× → (R/P)× be the canonical inclusion. If p splits in O, let ξ¯: (O/p̄)× →
(R/P)× denote precomposition of ξ with complex conjugation.
Remark 5.2.5. If µ ∈ Z has reduction ν ∈ Zp modulo P, then ν1 (x) = x−1 mod P and
ν2 (x) = x mod P for x ≡ 1 mod N with x prime to p. From this, we obtain the following.

i. If p does not divide f1 (resp., f2 ) for some choice of µ ∈ Z reducing to ν, then

ν1 |(O/p)× = ξ −1 (resp., ν2 |(O/p)× = ξ).

ii. Suppose that p splits in O. If p̄ does not divide fk for some k ∈ {1, 2} and some
(equivalently, all) µ reducing to ν then νk |(O/p̄)× = 1.

The converse to (i) (resp., (ii)) holds if N is divisible by at most a single power of p (resp., p̄).

Lemma 5.2.6. Suppose that µ ∈ Z is such that µ1 µ2 is primitive at some prime q dividing N .
Then q cannot divide both f1 and f2 .

Proof. The conductor of µ1 µ2 divides f1 ∩ f2 , so if neither µ1 nor µ2 is primitive at q, then

µ1 µ2 cannot be either. But if one of µ1 or µ2 is primitive at q, the fact that f1 f2 divides N
forces the other to have conductor prime to q.

Note that Lemma 5.2.6 implies that the first homology of the Borel-Serre boundary vanishes
in prime level, as there is no Hecke character of type (±1, 0) and trivial conductor. This can
also be seen using the fact that, in this case, each Borel subgroup contains an involution which
acts by conjugation as −1 on the unipotent subgroup.

42
Lemma 5.2.7. Let µ ∈ Z reduce to ν ∈ Zp modulo P.
a. If µ1 µ2 is primitive at p, then either ν1 |(O/p)× = ξ −1 or ν2 |(O/p)× = ξ.

b. Suppose that p is split in O. If µ1 µ2 is primitive at p̄, then either ν1 |(O/p̄)× = 1 or

ν2 |(O/p̄)× = 1.
Moreover, we need not assume the primitivity of µ1 µ2 in (a) (resp., (b)) if N is divisible by at
most a single power of p (resp., p̄).
Proof. If the conclusion of part (a) (resp., (b)) fails for ν, then p (resp., p̄) divides both f1 and
f2 by Remark 5.2.5. Then Lemma 5.2.6 tells us that µ1 µ2 cannot be primitive at p (resp., p̄). If
N is not divisible by the square of p (resp., p̄), the assumed failure already gives contradiction
of the final statement by the final statement of Remark 5.2.5.

Let I denote the Eisenstein ideal of the Hecke algebra H = HZ′ (N ) generated by the
elements Tq − (N q + Sq ) for primes q ∤ N . For an H-module M , let us write MEis to denote
the direct sum of the localizations of M at the maximal ideals of H containing I.
Let χ : Q′ → Oχ× be a character of the prime-to-p part Q′ of ClN (F ). The conductor of χ,
extended to ClN (F ) by the trivial character, then automatically has p-part dividing the product
of the primes of O over p.
Proposition 5.2.8. Suppose that N is divisible by at most one power of each prime over p, as
well as the following:
i. If p is split in O, then χ|(O/p)× ̸≡ ξ mod P and χ|(O/p̄)× ̸≡ ξ¯ mod P.

ii. If p is inert in O, then χ|(O/p)× ̸≡ ξ, ξ p mod P.

iii. If p is ramified in O, then χ|(O/p)× ̸≡ ξ mod P.

(χ)
Then the group H 1 (∂X1BS (N ), Zp )Eis is trivial.
Proof. By Proposition 5.2.4, it suffices to show that if ν ∈ Zp with ν1 ν2 = χ mod P, then
there exists some q not dividing N p such that ω(q) + ν1 ν2 (q) and ων1 (q) + ν2 (q) differ in
R/P. If this fails, then linear independence of characters forces ν1 = 1 or ν2 = ω. If ν1 = 1,
then part (a) of Lemma 5.2.7 (in its stronger form given by the final statement of the lemma)
tells us that ν2 |(O/p)× = ξ, so χ2 |(O/p)× ≡ ξ mod P.
Now suppose that ν2 = ω. Since ω(x) = xx̄ for x ∈ O prime to N p, the reduction of
ω|(O/p̄)× modulo P equals ξ¯ if p is split, ξ 2 is p is ramified and ξ p+1 if p is inert. If p is split,
part (b) of Lemma 5.2.7 then forces ν1 |(O/p̄)× = 1, whereas if p is inert or ramified, part (a) of
said lemma tells us that ν1 |(O/p)× = ξ −1 . Putting this together, we obtain that χ|(O/p̄)× modulo
P is ξ¯ if p is split, ξ if p is ramified, and ξ p if p is inert.

43
 Identifying ClN (F ) with Gal(F (N )/F ) via the Artin map R, we may conclude the
following.

Theorem 5.2.9. Let N , p ≥ 7, and χ ̸= ω 2 also satisfy the conditions of Proposition 5.2.8.
Then we have a homomorphism

ΠχN : H1 (X1 (N ), Zp )(χ) → (K2 (O′ (N )) ⊗ Zp )(χ) ,

compatible with the likewise denoted map on H1 (Y1 (N ), Zp )(χ) defined after (5.2), which is
Eisenstein away from N in the sense of (5.3) and Proposition 5.1.3.

Remark 5.2.10. We expect that ΠχN is Eisenstein at primes dividing the level as well. Here,
this would mean that ΠχN ◦ Uq∗ = ΠχN for each q dividing N , where Uq∗ is a dual Hecke operator
attached to q. This should allow one to remove the condition that N is divisible by at most one
power of p in Theorem 5.2.9. That is, though we have not written down details, the argument of
Proposition 5.2.8 for the Eisenstein localization of boundary homology including each Up∗ − 1
for p dividing p should go through with the same list of characters, as Up∗ should act as 0 on
H 1 (∂X1BS (N ), C)µ in the case that µ1 and µ2 are both ramified at p.

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Emmanuel Lecouturier Romyar Sharifi

Institute for Theoretical Sciences Department of Mathematics
Westlake University University of California, Los Angeles
No. 600 Dunyu Road, Sandun town, Xihu district 520 Portola Plaza
310030 Hangzhou, Zhejiang, China Los Angeles, CA 90095, USA
elecoutu@westlake.edu.cn sharifi@math.ucla.edu

Sheng-Chi Shih Jun Wang

Faculty of Mathematics Institute for Advanced Study in Mathematics
University of Vienna Harbin Institute of Technology
Oskar-Morgenstern-Platz 1 No. 92 West Da Zhi Street, Nan Gang District
A-1090 Wien, Austria 150001 Harbin, Heilongjiang, China
shengchishih@gmail.com junwangmath@hit.edu.cn
46