Symplectic constructions for `-adic local systems

George Pappas

Michigan State University

AGQFT
May 27, 2020

G. Pappas May 27, 2020 1 / 23

“Arithmetic topology” analogies

Weil, ... Mazur, ... Kim, ...

3-manifold ←→ integers of a number field ←→ curve over a finite field

2-manifold ←→ local (p-adic) field ←→ curve over an algebraically closed field k̄

1-manifold ←→ finite field ←→ Laurent power series k̄((t))

G. Pappas May 27, 2020 2 / 23

(A) X algebraic curve (smooth, projective) defined over the field k
k̄ = algebraic closure of k, Gk = Gal(k̄/k)
X̄ = X ⊗k k̄ the corresponding geometric curve.

1 → π1 (X̄) → π1 (X) → Gk → 1.

π1 (X), π1 (X̄) = Grothendieck’s étale fundamental groups (omit base point).

k = Fq finite field, Gk = Ẑ := limn Z/nZ, topologically generated by
←−
Frobenius φ : a 7→ aq .
Choose lift φ̃ ∈ π1 (X), γ 7→ φ̃γ φ̃−1 is an automorphism of π1 (X̄);
Its class in Out(π1 (X̄)) is independent of the lift.

G. Pappas May 27, 2020 3 / 23

(B) S = Riemann surface; Γ = π1 (S), MCG(S) = Mapping class group of S,
MCG(S) ' Out+ (Γ) ⊂ Out(Γ) of index 2.
Pick σ ∈ Out+ (Γ) ' MCG(S), represented by σ : S → S.
Mapping torus:
Mσ := S × [0, 1]/ ∼, (x, 0) ∼ (σ(x), 1),
is a 3-manifold fibered over the circle Mσ → S 1 with fiber S.

1 → Γ = π1 (S) → π1 (Mσ ) → Z → 1

G. Pappas May 27, 2020 4 / 23

 R
Regulator/Cheeger-Chern-Simons class: H3 (SLn (C), Z) −−−→ C/4π 2 Z ' iR ⊕ R/ Z .
(Beilinson regulator: K3 (Q̄) → C/(2πi)2 Z.)
Now suppose M is a 3-manifold (closed, oriented), ρ : π1 (M ) → SLn (C).
H3 (ρ) R
− H3 (π1 (M ), Z) −−−−−→ H3 (SLn (C), Z) −−−→ C/4π 2 Z ' iR ⊕ R/ Z .
Z→

Z
i 2
VolC (ρ) := −i Image(1) = − s∗ (Tr(A ∧ dA + A ∧ A ∧ A)).
2 M 3

Write: VolC (ρ) = Vol(ρ) + iCS(ρ).

If M has hyperbolic structure given by ρ : π1 (M ) → SL2 (C), then

Vol(ρ) = Vol(M ), CS(ρ) = Chern-Simons invariant of M.

G. Pappas May 27, 2020 5 / 23

Character “variety” of Γ = π1 (S):

Charn (Γ) = Hom(Γ, SLn (C))/SLn (C).

Tangent space of Char(Γ) at [ρ]

T[ρ] ∼
= H 1 (Γ, Ad0 (ρ)).

Pairing:
∪ κ
H 1 (Γ, Ad0 (ρ)) × H 1 (Γ, Ad0 (ρ)) − → H 2 (Γ, C) ∼
→ H 2 (Γ, Ad0 (ρ) ⊗C Ad0 (ρ)) − =C

κ : X ⊗ Y 7→ Trace(XY )
gives T[ρ] × T[ρ] → C; this defines ω ∈ ∧2 ΩChar(Γ) = Ω2Char(Γ) .
Goldman:
ω is closed (dω = 0) and non-degenerate, so
ω defines a canonical (complex) symplectic structure on Char(Γ).
G. Pappas May 27, 2020 6 / 23
The mapping class group MCG(S) ∼
= Out+ (Γ) acts on Char(Γ) by symplectomorphisms:

σ ∗ (ω) = ω, ∀σ ∈ MCG(S).

For σ ∈ MCG(S), recall the mapping torus Mσ . Consider:

Char(Mσ ) := Hom(π1 (Mσ ), SLn (C))/SLn (C) = Char(Γ)σ=id

This is the intersection of two Lagrangians:

Char(Mσ ) = ∆ ∩ Graph(σ) ⊂ Char(Γ) × Char(Γ).

(Pantev-Toen-Vaquié-Vezzosi:)
Char(Mσ ) has a canonical “(−1)-shifted symplectic structure”,

G. Pappas May 27, 2020 7 / 23

(Joyce, Brav-Bussi-Joyce:) Locally, Char(Mσ ) is the critical locus of a function Cσ :

Char(Mσ ) = {x | dCσ (x) = 0}

(Morally, Cσ is given by the Chern-Simons functional on connections on SLn (C)-bundles over

Mσ .)

Roughly: The vanishing cycles of Cσ should give perverse sheaves over Char(Mσ ) whose
homology should produce Floer homology groups for M = Mσ (cf. Abouzaid-Manolescu.) .

Note: These should be SLn (C)-Floer groups (“simpler” than SU(n)). Abouzaid-Manolescu
define such Floer groups from a pair of Lagrangians obtained from a Heegaard splitting of M .

G. Pappas May 27, 2020 8 / 23

Goal: Extend some of these constructions to “arithmetic cases”.

Fix a prime ` (odd), consider étale `-adic-local systems, or equivalently, `-adic representations,
i.e. continuous homomorphisms

Galois group, étale fundamental group → GLn (Q` ).

Q
Here, Galois group= lim(Finite Groups) ⊂ (Finite Groups) has the profinite topology,
←−
GLn (Q` ) has the topology induced by the `-adic absolute value on Q` .
Local system over a 3-dim object: A number field K or a curve X and

ρ : Gal(K̄/K) → GLn (Q` ) or, ρ : π1 (X) → GLn (Q` ).

Restrict to 2-dimensional objects: Local fields Kv , v a place of K, or the geometric curve X̄:
Consider ρ restricted to Gal(K̄v /Kv ) or, respectively, π1 (X̄).

G. Pappas May 27, 2020 9 / 23

The analogue of character varieties would be some type of “moduli spaces” for `-adic local
systems or `-adic Galois representations.
In general, we don’t have such good global spaces.
We usually access these moduli by considering (formal) deformations of a fixed Galois
representation or various rigid analytic universal families.

G. Pappas May 27, 2020 10 / 23

There are some attempts to build global moduli spaces that work under various assumptions.
They produce formal or rigid-analytic moduli stacks.

1) Consider “pseudo-representations” or “pseudo-characters” as substitutes for equivalence

classes of representations, sometimes up to semi-simplification (Chenevier,
C. Wang-Erickson).

2) For `-adic representations of Gal(Q̄` /Q` ) one can use `-adic Hodge theory
(Emerton-Gee).

Often, the correct approach is to use derived algebraic geometry (Galatius-Venkatesh).

Here we take a simple-minded approach and only look at (unobstructed) deformations.

G. Pappas May 27, 2020 11 / 23

To fix ideas consider a “function field case”:
k field of characteristic 6= `,
X smooth projective algebraic curve over k, X̄ = X ⊗k k̄.

ρ : π1 (X) → GLn (Z` )

continuous.

G. Pappas May 27, 2020 12 / 23

`-adic volume: `-adic Borel regulator:
`R
H3 (GLn (Z` ), Z` ) −→ Q`
corresponds to
3
R` ∈ Hcts (GLn (Z` ), Q` ) = limt (lims H 3 (GLn (Z/`s ), Z/`t )) ⊗Z` Q` .
←− −→
Now consider
3 ρ
Hcts → H 3 (X, Q` ) → H 1 (k, Q` (−1)),
(GLn (Z` ), Q` ) −
Vol` (ρ) := Image of R` ∈ H 1 (k, Q` (−1)).

k a finite field, or a local field of residue characteristic 6= `, H 1 (k, Q` (−1)) = 0.

[k : Q` ] = d, H 1 (k, Q` (−1)) ' Qd` .
k number field with r1 real and r2 complex places, conjecturally (Schneider),
H 1 (k, Q` (−1)) ' Qr` 1 +r2 .

G. Pappas May 27, 2020 13 / 23

Deformation spaces:
Start with
ρ0 : π1 (X) → SLn (F` ).
Set ρ̄0 := ρ0 |π1 (X̄) : π1 (X̄) → SLn (F` ). Assume ρ̄0 is geometrically irreducible.
There exists a universal formal deformation of ρ̄0 :

ρ̄uni : π1 (X̄) → SLn (Auni )

where Auni ' Z` [[t1 , . . . , tm ]].

DQ` = rigid `-adic analytic fiber of the formal scheme D given by Auni ' Z` [[t1 , . . . , tm ]]:

DQ` ' {(x1 , . . . , xm ) | |xi |` < 1, ∀i}

(| |` = `-adic absolute value, |`a |` = (1/`)a .)

G. Pappas May 27, 2020 14 / 23

Symplectic structure: A complete local ring. Milnor K2 (A) is generated by Steinberg
symbols {f, g}, with f , g ∈ A∗ .
df dg
d log : K2 (A) → Ω2A , d log({f, g}) = ∧ .
f g
Steinberg (central) extension

1 → K2 (A) → St(A) → SL(A) → 1.

ρ : Γ → SLn (A) gives [ρ] ∈ H 2 (Γ, K2 (A)). Consider

∩ d log
H 2 (Γ, K2 (A)) × H2 (Γ, Z) −
→ K2 (A) −−−→ Ω2A

If Z ' H2 (Γ, Z), 1 7→ c, we can set

ω = d log([ρ] ∩ c) ∈ Ω2A

Observe dω = d((d log)([ρ] ∩ c)) = 0, i.e. ω is closed.

G. Pappas May 27, 2020 15 / 23
A continuous version applies to ρ̄uni : π1 (X̄) → SLn (Auni ): Get

ω ∈ Ω2Auni := limn Ω2(Auni /mn )/Z`

←−

ω is closed,
ω is “perfect”,
ω ⊗Auni Auni /m2 ∈ ∧2 (m/m2 ) is given by Poincare duality on

(m/m2 )∗ = HomF` (m/m2 , F` ) ' H 1 (π1 (X̄), Ad0 (ρ̄0 ))

(just as Goldman’s form.)

ω is the canonical symplectic structure on D.

G. Pappas May 27, 2020 16 / 23

Galois action: Recall
1 → π1 (X̄) → π1 (X) → Gk → 1.
The representation ρ̄0 extends to ρ0 : π1 (X) → SLn (F` ):
Hence, the Galois group Gk acts on the formal deformation space D and its generic fiber DQ` .
On points: σ([ρ]) = [ρ · Innσ̃ ], where Innσ̃ = the automorphism of π1 (X) obtained by
conjugating by a lift σ̃ of σ in π1 (X).
For all σ ∈ Gk ,
σ(ω) = χ(σ)−1 · ω,
χ(σ)
where χ : Gk → Z` is the cyclotomic character, σ(ζ`m ) = ζ`m . In particular, if k = Fq ,
φ = Frobq , φ(ω) = q −1 ω.
Under some additional assumptions, the fixed points of the action of Gk on DQ`
correspond to ρ : π1 (X) → SLn (E), E/Q` finite, with restriction ρ̄ to π1 (X̄) that
deforms ρ̄0 .

G. Pappas May 27, 2020 17 / 23

Galois flow:
Set N = #Aut(Auni /m2 ).
Theorem: We can write
[
DQ` = Bc Bc ' {(x1 , . . . , xm ) | |xi |` ≤ (1/`)1/c },
c≥1

such that: For each c ≥ 1 there is (c) > 0 and a rigid analytic map

{t | |t|` ≤ (c)} × Bc → Bc , (t, x1 , . . . , xm ) 7→ ψ t (x1 , . . . , xm )

with
0 0
ψ t+t = ψ t · ψ t , i.e. is a flow in t,
interpolates the Galois action σ tN , for t ∈ Z, |t|` ≤ (c).

Proof: Interpolation of iterates (Poonen, Bell-Ghioca-Tucker; Litt).

G. Pappas May 27, 2020 18 / 23
 dψ t
Xσ = N −1 |t=0 , a rigid analytic vector field on DQ` ,
dt
µσ = Xσ ¬ ω, a rigid analytic 1-form on DQ` .

Basic identity: dµσ = − log` (χ(σ))ω (log` : Z∗` → Z` the standard `-adic logarithm.)
If χ(σ) = 1, then dµσ = 0, so there is a rigid analytic function Vσ on DQ` with

µσ = dVσ .

If x = (x1 , . . . , xm ) ∈ DQ` (E) corresponds to a representation which extends to the

arithmetic fundamental group,

ρ : π1 (X) → SLn (E)

then x is a critical point of Vσ , for all σ with χ(σ) = 1,

G. Pappas May 27, 2020 19 / 23
Consider
Q` [[u]] → O
bD ;
Q` ,x
u 7→ Vσ − Vσ (x).
The rigid Milnor fiber of Vσ at x:

M (x, σ) := Spf(O
bD ⊗
Q` ,x
ˆ Q` Q̄` )[1/u]

Berkovich: The étale cohomology (vanishing cycles)

i ˆ Q̄ ((u))) Q` ((u)), Q` )
Het (M (x, σ)⊗ `

make sense and is a finite dimensional vector space. We can set

X
λ(x, σ) = (−1)m (1 − (−1)i dimQ` Heti ˆ Q̄ ((u))) Q` ((u)), Q` )).
(M (x, σ)⊗ `
i

(analogue of the “Behrend function”.)

G. Pappas May 27, 2020 20 / 23

Unfortunately, this excludes the case k = finite field Fq .
Indeed, for φ = Frobq ∈ Gal(k̄/k), we have

φ(ω) = q −1 ω, dµφ = − log` (q) · ω

but q 6= 1 (!).
Idea (?) Find a symplectic potential λ, i.e. a 1-form λ with dλ = ω, which is “independent
of the curve X”.
Then log` (q)λ + µφ is closed, so

log` (q)λ + µφ = dVφ

for some Vφ .
The representations of the arithmetic fundamental group give x with

dVφ (x) = log` (q)λ(x).

G. Pappas May 27, 2020 21 / 23

If x ∈ D(OE ) corresponds to a representation ρx of the arithmetic fundamental group, then
Vol` (ρx ) makes sense.
What is its relation with Vσ (x)?
Z x0
Vol` (ρx ) − Vol` (ρx0 ) = Vσ (x) − Vσ (x0 ) = µσ ?
x

maybe up to constants?

G. Pappas May 27, 2020 22 / 23

 Thank you!

G. Pappas May 27, 2020 23 / 23