Volume and symplectic structure for ℓ-adic local systems

Abstract

We introduce a notion of volume for an ℓ-adic local system over an algebraic curve and, under some conditions, give a symplectic form on the rigid analytic deformation space of the corresponding geometric local system. These constructions can be viewed as arithmetic analogues of the volume and the Chern-Simons invariants of a representation of the fundamental group of a 3-manifold which fibers over the circle and of the symplectic form on the character varieties of a Riemann surface. We show that the absolute Galois group acts on the deformation space by conformal symplectomorphisms which extend to an ℓ-adic analytic flow. We also prove that the locus of local systems which are arithmetic over a cyclotomic extension is the critical set of a collection of rigid analytic functions. The vanishing cycles of these functions give additional invariants.

Introduction

In this paper, we introduce certain constructions for étale ${\mathbb{Z}}_{\ell }$-local systems (i.e. lisse ${\mathbb{Z}}_{\ell }$-sheaves) on proper algebraic curves defined a field of characteristic different from ℓ. In particular, using an ℓ-adic regulator, we define a notion of ℓ-adic volume. We also give a symplectic form on the (formal) deformation space of a modular representation of the geometric étale fundamental group of the curve. (In what follows, we will use the essentially equivalent language of ℓ-adic representations of the étale fundamental group.)

Our definitions can be viewed as giving analogues of constructions in the symplectic theory of character varieties of a Riemann surface and of the volume and the Chern-Simons invariants of representations of the fundamental group of a 3-manifold which fibers over $S^1$.

Let us recall some of these classical constructions, very briefly. We start with the symplectic structure on the character varieties of the fundamental group $\mathrm{\Gamma }={\pi }_1(\mathrm{\Sigma })$ of a (closed oriented) topological surface Σ. To fix ideas we consider the space

$$X_G(\mathrm{\Gamma })=\mathrm{Hom}(\mathrm{\Gamma },G)/G$$

parametrizing equivalence classes of representations of Γ with values in a connected real reductive group G; there are versions for complex reductive groups. (Here, we are being intentionally vague about the precise meaning of the quotient; what is clear is that it is taken for the conjugation action on the target.) Suppose that $\rho :\mathrm{\Gamma }\to G$ is a representation which gives a point $[\rho ]\in X_G(\mathrm{\Gamma })$. The tangent space $T_{[\rho ]}$ of $X_G(\mathrm{\Gamma })$ at $[\rho ]$ can be identified with ${\mathrm{H}}^1(\mathrm{\Gamma },{\mathrm{Ad}}_{\rho })$, where ${\mathrm{Ad}}_{\rho }$ is the Lie algebra of G with the adjoint action. Consider the composition

$${\mathrm{H}}^1(\mathrm{\Gamma },{\mathrm{Ad}}_{\rho })\times {\mathrm{H}}^1(\mathrm{\Gamma },{\mathrm{Ad}}_{\rho })\stackrel{\cup }{\to }{\mathrm{H}}^2(\mathrm{\Gamma },{\mathrm{Ad}}_{\rho }{\otimes }_{\mathbb{R}}{\mathrm{Ad}}_{\rho })\stackrel{B}{\to }{\mathrm{H}}^2(\mathrm{\Gamma },\mathbb{R})\cong \mathbb{R},$$

where B is induced by the Killing form and the last isomorphism is given by Poincare duality. This defines a non-degenerate alternating form

$$T_{[\rho ]}{\otimes }_{\mathbb{R}}{T}_{[\rho ]}\to \mathbb{R},$$

i.e. ${\omega }_{[\rho ]}\in {\wedge }^2{T}_{[\rho ]}^{⁎}$. By varying ρ we obtain a 2-form ω over $X_G(\mathrm{\Gamma })$. Goldman [25] shows that this form is closed, i.e. $d\omega =0$, and so it gives a symplectic structure (at least over the space of “good” ρ's which is a manifold). Note here that the mapping class group of the surface Σ acts naturally on the character variety by symplectomorphisms, i.e. maps that respect Goldman's symplectic form. In turn, this action relates to many fascinating mathematical structures.

Next, we discuss the notion of a volume of a representation. Here, again to fix ideas, we start with a (closed oriented) smooth 3-manifold M and take ${\mathrm{\Gamma }}_0={\pi }_1(M)$. Let $X=G/H$ be a contractible G-homogenous space of dimension 3 and choose a G-invariant volume form ${\omega }_X$ on X. A representation $\rho :{\mathrm{\Gamma }}_0={\pi }_1(M)\to G$ gives a flat X-bundle space $\pi :\tilde{M}\to M$ with G-action. The volume form ${\omega }_X$ naturally induces a 3-form ${\omega }_X^{\rho }$ on $\tilde{M}$. Take $s:M\to \tilde{M}$ to be a differentiable section of π and set

$$\mathrm{Vol}(\rho )=\underset{M}{\int }{s}^{⁎}{\omega }_X^{\rho }\in \mathbb{R}$$

which can be seen to be independent of the choice of section s ([24]). The map $\rho \mapsto \mathrm{Vol}(\rho )$ gives an interesting real-valued function on the space $X_G({\mathrm{\Gamma }}_0)$.

An important special case is when M is hyperbolic, $G={\mathrm{PSL}}_2(\mathbb{C})$, $X={\mathbb{H}}^3=\mathbb{C}\times {\mathbb{R}}_{>0}$ (hyperbolic 3-space), ${\omega }_X$ is the standard volume form on ${\mathbb{H}}^3$, and ${\rho }_{\mathrm{hyp}}:{\pi }_1(M)\to {\mathrm{PSL}}_2(\mathbb{C})={\mathrm{Isom}}^{+}({\mathbb{H}}^3)$ is the representation associated to the hyperbolic structure of $M={\mathbb{H}}^3/{\mathrm{\Gamma }}_0$. Then, $\mathrm{Vol}({\rho }_{\mathrm{hyp}})=\mathrm{Vol}(M)$, the hyperbolic volume of M. The Chern-Simons invariant $\mathrm{CS}(M)$ of M is also related. For this, compose the map

$${\mathrm{H}}_3(M,\mathbb{Z})={\mathrm{H}}_3({\pi }_1(M),\mathbb{Z})\stackrel{{\rho }_{\mathrm{hyp}}}{\to }{\mathrm{H}}_3({\mathrm{PSL}}_2(\mathbb{C}),\mathbb{Z})$$

with the “regulator”

$$\mathfrak{R}:{\mathrm{H}}_3({\mathrm{PSL}}_2(\mathbb{C}),\mathbb{Z})\to \mathbb{C}/{\pi }^2\mathbb{Z}.$$

The product of −i with the image of the fundamental class $[M]$ under this composition is the “complex volume”

$${\mathrm{Vol}}_{\mathbb{C}}({\rho }_{\mathrm{hyp}})={\mathrm{Vol}}_{\mathbb{C}}(M)=\mathrm{Vol}(M)+i2{\pi }^2\mathrm{CS}(M)$$

([45]); this can also be given by an integral over M.

A straightforward generalization of this construction leads to the definition of a complex volume ${\mathrm{Vol}}_{\mathbb{C}}(\rho )$ for representations $\rho :{\pi }_1(M)\to {\mathrm{SL}}_n(\mathbb{C})$ (see, for example, [23]). This uses the regulator maps (universal Cheeger-Chern-Simons classes)

$${\mathfrak{R}}_n:{\mathrm{H}}_3({\mathrm{SL}}_n(\mathbb{C}),\mathbb{Z})\to \mathbb{C}/{(2\pi i)}^2\mathbb{Z}.$$

(See also [16] and [26].)

We now return to the arithmetic set-up of local systems over algebraic curves. Recall that we are considering formal deformations of a modular (modulo ℓ) representation. We show that when the modular representation is the restriction of a representation of the arithmetic étale fundamental group, the absolute Galois group acts on the deformation space by “conformal symplectomorphisms” (i.e. scaling the symplectic form) which extend to an ℓ-adic analytic flow. This gives an analogue of the action of the mapping class group on the character variety by symplectomorphisms we mentioned above. We show that if the curve is defined over a field k, the action of a Galois automorphism that fixes the field extension $k({\zeta }_{{\ell }^{\infty }})$ generated by all ℓ-power roots of unity, is “Hamiltonian”. We use this to express the set of deformed representations that extend to a representation of a larger fundamental group over $k({\zeta }_{{\ell }^{\infty }})$ as the intersection of the critical loci for a set of rigid analytic functions $V_{\sigma }$, where σ ranges over $\mathrm{Gal}(k^{\mathrm{sep}}/k({\zeta }_{{\ell }^{\infty }}))$. The Milnor fibers and vanishing cycles of $V_{\sigma }$ provide interesting constructions.

Let us now explain this in more detail. Let ℓ be a prime which we assume is odd, for simplicity. Suppose that X is a smooth geometrically connected proper curve over a field k of characteristic prime to ℓ. The properness of the curve is quite important for most of the constructions. We fix an algebraic closure $\overline{k}$. Denote by $G_k=\mathrm{Gal}(k^{\mathrm{sep}}/k)$ the Galois group where $k^{\mathrm{sep}}$ is the separable closure of k in $\overline{k}$, by $k({\zeta }_{{\ell }^{\infty }})={\cup }_{n}k({\zeta }_{{\ell }^n})$ the subfield of $k^{\mathrm{sep}}$ generated over k by all the ${\ell }^n$-th roots of unity and by ${\chi }_{\mathrm{cycl}}:G_k\to {\mathbb{Z}}_{\ell }^{⁎}$ the cyclotomic character. Fix a $\overline{k}$-point $\overline{x}$ of X, and consider the étale fundamental groups which fit in the canonical exact sequence

$$1\to {\pi }_1(X{\times }_k\overline{k},\overline{x})\to {\pi }_1(X,\overline{x})\to G_k\to 1.$$

For simplicity, we set $\overline{X}=X{\times }_k\overline{k}$ and omit the base point $\overline{x}$. Let $\mathcal{F}$ be an étale ${\mathbb{Z}}_{\ell }$-local system of rank $d>1$ over X. The local system $\mathcal{F}$ corresponds to a continuous representation $\rho :{\pi }_1(X)\to {\mathrm{GL}}_d({\mathbb{Z}}_{\ell })$.

The ℓ-adic volume $\mathrm{Vol}(\mathcal{F})$ of $\mathcal{F}$ is, by definition, a continuous cohomology class

$$\mathrm{Vol}(\mathcal{F})\in {\mathrm{H}}^1(k,{\mathbb{Q}}_{\ell }(-1)).$$

Here, as usual, ${\mathbb{Q}}_{\ell }(n)={\mathbb{Q}}_{\ell }{\otimes }_{{\mathbb{Z}}_{\ell }}{\chi }_{\mathrm{cycl}}^n$ is the n-th Tate twist.

Note that if k is a finite field or a finite extension of ${\mathbb{Q}}_p$ with $p\ne \ell$, then we have ${\mathrm{H}}^1(k,{\mathbb{Q}}_{\ell }(-1))=(0)$. If k is a finite extension of ${\mathbb{Q}}_{\ell }$, then

$${\mathrm{H}}^1(k,{\mathbb{Q}}_{\ell }(-1))\simeq {\mathbb{Q}}_{\ell }^{[k:{\mathbb{Q}}_{\ell }]}.$$

If k is a number field with $r_1$ real and $r_2$ complex places, then assuming a conjecture of Schneider [53], we have

$${\mathrm{H}}^1(k,{\mathbb{Q}}_{\ell }(-1))\simeq {\mathbb{Q}}_{\ell }^{r_1+r_2}.$$

In fact, using the restriction-inflation exact sequence, we can give $\mathrm{Vol}(\mathcal{F})$ as a continuous homomorphism

$$\mathrm{Vol}(\mathcal{F}):\mathrm{Gal}(k^{\mathrm{sep}}/k({\zeta }_{{\ell }^{\infty }}))\to {\mathbb{Q}}_{\ell }(-1)$$

which is equivariant for the action of $\mathrm{Gal}(k({\zeta }_{{\ell }^{\infty }})/k)$.

To define $\mathrm{Vol}(\mathcal{F})$ we use a continuous 3-cocycle

$${\mathfrak{r}}_{{\mathbb{Z}}_{\ell }}:{\mathbb{Z}}_{\ell }〚{\mathrm{GL}}_d{({\mathbb{Z}}_{\ell })}^3〛\stackrel{}{\to }{\mathbb{Q}}_{\ell }$$

that corresponds to the ℓ-adic Borel regulator [31]. The quickest way is probably as follows (but see also §4.4): Using the Leray-Serre spectral sequence we obtain a homomorphism

$${\mathrm{H}}^3({\pi }_1(X),{\mathbb{Q}}_{\ell }/{\mathbb{Z}}_{\ell })\to {\mathrm{H}}_{\mathrm{\text{ét}}}^3(X,{\mathbb{Q}}_{\ell }/{\mathbb{Z}}_{\ell })\to {\mathrm{H}}^1(k,{\mathrm{H}}_{\mathrm{\text{ét}}}^2(\overline{X},{\mathbb{Q}}_{\ell }/{\mathbb{Z}}_{\ell }))={\mathrm{H}}^1(k,{\mathbb{Q}}_{\ell }/{\mathbb{Z}}_{\ell }(-1)).$$

Taking Pontryagin duals gives

$${\mathrm{H}}^1{(k,{\mathbb{Q}}_{\ell }/{\mathbb{Z}}_{\ell }(-1))}^{⁎}={\mathrm{H}}_1(G_k,{\mathbb{Z}}_{\ell }(1))\to {\mathrm{H}}_3({\pi }_1(X),{\mathbb{Z}}_{\ell }).$$

Now compose this with the map given by ρ and the ℓ-adic regulator to obtain

$${\mathrm{H}}_1(G_k,{\mathbb{Z}}_{\ell }(1))\to {\mathrm{H}}_3({\pi }_1(X),{\mathbb{Z}}_{\ell })\stackrel{{\mathrm{H}}_3(\rho )}{\to }{\mathrm{H}}_3({\mathrm{GL}}_d({\mathbb{Z}}_{\ell }),{\mathbb{Z}}_{\ell })\stackrel{{\mathfrak{r}}_{{\mathbb{Z}}_{\ell }}}{\to }{\mathbb{Q}}_{\ell }.$$

This, by the universal coefficient theorem, gives $\mathrm{Vol}(\mathcal{F})\in {\mathrm{H}}^1(k,{\mathbb{Q}}_{\ell }(-1))$, up to sign. In fact, we give an “explicit” continuous 1-cocycle that represents $\mathrm{Vol}(\mathcal{F})$ by a construction inspired by classical Chern-Simons theory [20].

Let c be a fundamental 2-cycle in ${\mathbb{Z}}_{\ell }〚{\pi }_1{(\overline{X})}^2〛$. Lift $\sigma \in G_k$ to $\tilde{\sigma }\in {\pi }_1(X)$ and consider the (unique up to boundaries) 3-chain $\delta (\tilde{\sigma },c)\in {\mathbb{Z}}_{\ell }〚{\pi }_1{(\overline{X})}^3〛$ with boundary

$$\partial (\delta (\tilde{\sigma },c))=\tilde{\sigma }\cdot c\cdot {\tilde{\sigma }}^{-1}-{\chi }_{\mathrm{cycl}}(\sigma )\cdot c.$$

Also, let $F_{\rho (\tilde{\sigma })}(\rho (c))$ be the (continuous) 3-chain for ${\mathrm{GL}}_d({\mathbb{Z}}_{\ell })$ which gives the “canonical” boundary for the 2-cycle $\rho (\tilde{\sigma })\cdot \rho (c)\cdot \rho {(\tilde{\sigma })}^{-1}-\rho (c)$. We set

$$B(\sigma ):={\mathfrak{r}}_{{\mathbb{Z}}_{\ell }}[\rho (\delta (\tilde{\sigma },c))-F_{\rho (\tilde{\sigma })}(\rho (c))]\in {\mathbb{Q}}_{\ell }.$$

The map $G_k\to {\mathbb{Q}}_{\ell }(-1)$ given by $\sigma \mapsto {\chi }_{\mathrm{cycl}}{(\sigma )}^{-1}B(\sigma )$ is a continuous 1-cocycle which is independent of choices up to coboundaries and whose class is $\mathrm{Vol}(\mathcal{F})$.

This explicit construction is more flexible and can be applied to continuous representations $\overline{\rho }:{\pi }_1(\overline{X})\to {\mathrm{GL}}_d(A)$, where A is a more general ℓ-adic ring. In fact, we do not need that $\overline{\rho }$ extends to ${\pi }_1(X)$ but only that it has the following property: There is continuous homomorphism $\phi :G_k\to \mathrm{Aut}(A)$ such that, for every $\sigma \in G_k$, there is a lift $\tilde{\sigma }\in {\pi }_1(X)$, and a matrix $h_{\tilde{\sigma }}\in {\mathrm{GL}}_d(A)$, with

$$\overline{\rho }(\tilde{\sigma }\cdot \gamma \cdot {\tilde{\sigma }}^{-1})=h_{\tilde{\sigma }}\cdot \phi (\sigma )(\overline{\rho }(\gamma ))\cdot h_{\tilde{\sigma }}^{-1},\forall \gamma \in {\pi }_1(\overline{X}).$$

(In the above case, $A={\mathbb{Z}}_{\ell }$, $\phi (\sigma )=\mathrm{id}$, and $h_{\tilde{\sigma }}=\rho (\tilde{\sigma })$.) We again obtain a (continuous) class

$${\mathrm{Vol}}_{\rho ,\phi }\in {\mathrm{H}}^1(k,\mathcal{O}(\mathcal{D})(-1))$$

which is independent of choices. In this, $\mathcal{O}(\mathcal{D})$ is the ring of analytic functions on the rigid generic fiber $\mathcal{D}=\mathrm{Spf}(A)[1/\ell ]$ of A, with $G_k$-action given by φ.

In particular, with some more work (see §5.1, and especially Proposition 5.1.2), we find that this construction applies to the case that A is the universal formal deformation ring of an absolutely irreducible representation ${\overline{\rho }}_0:{\pi }_1(\overline{X})\to {\mathrm{GL}}_d({\mathbb{F}}_{\ell })$ which is the restriction of a continuous ${\rho }_0:{\pi }_1(X)\to {\mathrm{GL}}_d({\mathbb{F}}_{\ell })$. Then the Galois group $G_k$ acts on A and the action, by its definition, satisfies the condition above. In this case, the ring A is (non-canonically) a formal power series ring $A\simeq W({\mathbb{F}}_{\ell })〚x_1,\dots ,x_r〛$ and $\mathcal{O}(\mathcal{D})$ is the ring of rigid analytic functions on the open unit ℓ-adic polydisk.

Suppose now that ℓ is prime to d. We show that, in the above case of a universal formal deformation with determinant fixed to be a given character $ϵ:{\pi }_1(\overline{X})\to {\mathbb{Z}}_{\ell }^{⁎}$, the ring A carries a canonical “symplectic structure”. This is reminiscent of the canonical symplectic structure on the character varieties of the fundamental groups of surfaces [25]. Here, it is given by a continuous non-degenerate 2-form $\omega \in {\wedge }^2{\mathrm{\Omega }}_{A/W}^{\mathrm{ct}}$ which is closed, i.e. $d\omega =0$.

Let us explain our construction of ω. By definition,

$${\mathrm{\Omega }}_{A/W}^{\mathrm{ct}}={\underleftarrow{\lim }}_n{\mathrm{\Omega }}_{A/{\mathfrak{m}}^n/W},$$

where $\mathfrak{m}$ is the maximal ideal of A. Consider the map

$$d\log :{\mathrm{K}}_2^{\mathrm{ct}}(A)={\underleftarrow{\lim }}_n{\mathrm{K}}_2(A/{\mathfrak{m}}^n)\to {\wedge }^2{\mathrm{\Omega }}_{A/W}^{\mathrm{ct}}={\underleftarrow{\lim }}_n{\mathrm{\Omega }}_{A/{\mathfrak{m}}^n/W}$$

obtained as the limit of

$$d{\log }_{(n)}:{\mathrm{K}}_2(A/{\mathfrak{m}}^n)\to {\mathrm{\Omega }}_{A/{\mathfrak{m}}^n/W}.$$

We first define a (finer) invariant

$$\kappa =\underset{n}{\underleftarrow{\lim }}{\kappa }_n$$

of the universal formal deformation ${\rho }_A:{\pi }_1(\overline{X})\to {\mathrm{GL}}_d(A)$ with values in the limit ${\mathrm{K}}_2^{\mathrm{ct}}(A)={\underleftarrow{\lim }}_n{\mathrm{K}}_2(A/{\mathfrak{m}}^n)$. For $n\ge 1$, ${\kappa }_n$ is the image of 1 under the composition

$${\mathbb{Z}}_{\ell }(1)\stackrel{{\mathrm{tr}}^{⁎}}{\to }{\mathrm{H}}_2({\pi }_1(\overline{X}),{\mathbb{Z}}_{\ell })\to {\mathrm{H}}_2({\mathrm{SL}}_{d+1}(A/{\mathfrak{m}}^n),{\mathbb{Z}}_{\ell })\stackrel{\sim }{\to }{\mathrm{K}}_2(A/{\mathfrak{m}}^n).$$

Here the second map is induced by $\rho \oplus {ϵ}^{-1}$, and the third is the isomorphism given by stability and the Steinberg sequence

$$1\to {\mathrm{K}}_2(A/{\mathfrak{m}}^n)\to \mathrm{St}(A/{\mathfrak{m}}^n)\to \mathrm{SL}(A/{\mathfrak{m}}^n)\to 1.$$

Then the 2-form $\omega \in {\wedge }^2{\mathrm{\Omega }}_{A/W}^{\mathrm{ct}}$ is, by definition, the image

$$\omega =d\log (\kappa ).$$

The closedness of ω follows immediately since all the 2-forms in the image of dlog are closed. We show that the form ω is also given via cup product and Poincare duality, just as in the construction of Goldman's form above (see Theorem 5.5.2), and that is non-degenerate. This is done by examining the tangent space of the Steinberg extension using some classical work of van der Kallen.

In fact, this also provides an alternate argument for the closedness of Goldman's 2-form [25] on character varieties. Showing that this form (which is defined using cup product and duality) is closed, and thus gives a symplectic structure, has a long and interesting history. The first proof, by Goldman, used a gauge theoretic argument that goes back to Atiyah and Bott. A more direct proof using group cohomology was later given by Karshon [34]. Other authors gave different arguments that also extend to parabolic character varieties for surfaces with boundary, see for example [28]. The approach here differs substantially: We first define a 2-form which is easily seen to be closed using ${\mathrm{K}}_2$, and then we show that it agrees with the more standard form constructed using cup product and duality. Let us mention here that Pantev-Toen-Vaquié-Vezzosi have given in [47] a general approach for constructing symplectic structures on similar spaces (stacks) which uses derived algebraic geometry. In fact, following this, the existence of the canonical symplectic structure on $\mathrm{Spf}(A)[1/\ell ]$ was also shown, and in a greater generality, by Antonio [3], by extending the results of [47] to a rigid-analytic set-up. This uses, among other ingredients, the theory of derived rigid-analytic stacks developed in work of Porta-Yu [49] (see also [50]). Our argument is a lot more straightforward and, in addition, gives the symplectic form over the formal scheme $\mathrm{Spf}(A)$. (However, the derived approach would be important for handling the cases in which the representation is not irreducible.)

It is not hard to see (cf. [15]), that the automorphisms $\phi (\sigma )$ of A given by $\sigma \in G_k$, respect the form ω up to Tate twist:

$$\phi (\sigma )(\omega )={\chi }_{\mathrm{cycl}}^{-1}(\sigma )\omega .$$

(So they are “conformal symplectomorphisms” of a restricted type.) In particular, if k is a finite field of order $q=p^f$, prime to ℓ, and σ is the geometric Frobenius ${\mathrm{Frob}}_q$, the corresponding automorphism $\phi =\phi ({\mathrm{Frob}}_q)$ satisfies $\phi (\omega )=q\cdot \omega$.

The automorphism $\phi (\sigma )$ can be extended to give a “flow”: Using an argument of Poonen on interpolation of iterates, we show that we can write $\mathcal{D}$ as an increasing union of affinoids

$$\mathcal{D}={\bigcup }_{c\in \mathbb{N}}{\overline{\mathcal{D}}}_c$$

(each ${\overline{\mathcal{D}}}_c$ isomorphic to a closed ball of radius ${\ell }^{-1/c}$) such that the following is true:

There is $N\ge 1$, and for each c, there is a rational number $\epsilon (c)>0$, such that, for $\sigma \in G_k$, the action of ${\sigma }^{nN}$ on A interpolates to an ℓ-adic analytic flow ${\psi }^t:={\sigma }^{tN}$ on ${\overline{\mathcal{D}}}_c$, defined for $|t{|}_{\ell }\le \epsilon (c)$, i.e. to a rigid analytic map

$$\{t||t{|}_{\ell }\le \epsilon (c)\}\times {\overline{\mathcal{D}}}_c\to {\overline{\mathcal{D}}}_c,(t,\mathbf{x})\mapsto {\psi }^t(\mathbf{x}),$$

with ${\psi }^{t+t^{\prime }}={\psi }^t\cdot {\psi }^{t^{\prime }}$. As $c\mapsto +\infty$, $\epsilon (c)\mapsto 0$, and so we can think of this as a flow on $\mathcal{D}$ which, as we approach the boundary, only exists for smaller and smaller times. (A similar construction is given by Litt in [38] and, in the abelian case, by Esnault-Kerz [17].) We show that if ${\chi }_{\mathrm{cycl}}(\sigma )=1$, this flow is symplectic and in fact Hamiltonian, i.e. it preserves the level sets of an ℓ-adic analytic function $V_{\sigma }\in \mathcal{O}(A)$. More precisely, the flow ${\sigma }^{tN}$ gives a vector field $X_{\sigma }$ on $\mathcal{D}$ whose contraction with ω is the exact 1-form $d{V}_{\sigma }$. It follows that the critical points of the function $V_{\sigma }$ are fixed by the flow. We use this to deduce that the intersection of the critical loci of $V_{\sigma }$ corresponds to representations of ${\pi }_1(\overline{X})$ that extend to ${\pi }_1(X{\times }_k{k}^{\prime })$ for some finite extension $k^{\prime }$ of $k({\zeta }_{{\ell }^{\infty }})$. The flow ${\psi }^t$ is an interesting feature of the rigid deformation space $\mathcal{D}$ that we think deserves closer study. Versions of this flow construction have already been used in [38], [17], [18], to obtain results about the set of representations which extend to the arithmetic étale fundamental group. It remains to see if its symplectic nature, explained here, can provide additional information.

The inspiration for these constructions comes from a wonderful idea of M. Kim [36] (see also [13]) who, guided by the folkore analogy between 3-manifolds and rings of integers in number fields and between knots and primes, gave a construction of an arithmetic Chern-Simons invariant for finite gauge group. He also suggested ([37]) to look for more general Chern-Simons type theories in number theory that resemble the corresponding theories in topology and mathematical physics. An important ingredient of classical Chern-Simons theory is the symplectic structure on the character variety of a closed orientable surface: When the surface is the boundary of a 3-manifold, the Chern-Simons construction gives a section of a line bundle over the character variety. The line bundle has a connection with curvature given by Goldman's symplectic form. One can try to imitate this construction in number theory by regarding the 3-manifold with boundary as analogous to a ring of integers with a prime inverted.

In this paper, we have a different, simpler, analogy: Our topological model is a closed 3-manifold M fibering over the circle $S^1$ with fiber a closed orientable surface Σ of genus ≥1 with fundamental group $\mathrm{\Gamma }={\pi }_1(\mathrm{\Sigma })$. The monodromy gives an element σ of the mapping class group $\mathrm{Out}(\mathrm{\Gamma })$, so we can take M to be the “mapping torus” $\mathrm{\Sigma }\times [0,1]/\sim$, where $(a,0)\sim (\tilde{\sigma }(a),1)$ with $\tilde{\sigma }:\mathrm{\Sigma }\to \mathrm{\Sigma }$ representing σ. There is an exact sequence

$$1\to \mathrm{\Gamma }\to {\pi }_1(M)\to \mathbb{Z}={\pi }_1(S^1)\to 1$$

and conjugation by a lift of $1\in \mathbb{Z}={\pi }_1(S^1)$ to ${\pi }_1(M)$ induces $\sigma \in \mathrm{Out}(\mathrm{\Gamma })$. A smooth projective curve X over the finite field $k={\mathbb{F}}_q$ is the analogue of M; in the analogy, $\overline{X}$ corresponds to Σ and the outer action of Frobenius on ${\pi }_1(\overline{X})$ to σ. The formalism extends to general fields k with the Galois group $G_k$ replacing ${\pi }_1(S^1)$. The ℓ-adic volume $\mathrm{Vol}(\mathcal{F})$ of a local system $\mathcal{F}$ on X corresponds to the (complex) volume of a representation of ${\pi }_1(M)$; this invariant includes the Chern-Simons invariant of the representation. Note that a representation of Γ gives a bundle with flat connection over Σ. This extends to a bundle with connection on M which corresponds to a representation of ${\pi }_1(M)$ if the connection is flat. Flatness occurs at critical points of the Chern-Simons functional. So, in our picture, $V_{\sigma }$ is an analogue of this functional. In fact, it is reasonable to speculate that the value $V_{\sigma }(\mathbf{x})$ at a point x which corresponds to a representation ρ of ${\pi }_1(X)$ relates to the ℓ-adic volume $\mathrm{Vol}(\rho )$; we have not been able to show such a statement.

In topology, such constructions are often a first step in the development of various “Floer-type” theories. It seems that the most relevant for our analogy are theories for non-compact complex groups like ${\mathrm{SL}}_2(\mathbb{C})$, for which there is a more algebraic treatment. A modern viewpoint for a particular version of these is, roughly, as follows: Since the character variety of Γ has a (complex, or even algebraic) symplectic structure and σ acts by a symplectomorphism, the fixed point locus of σ (which are points extending to representations of ${\pi }_1(M)$) is an intersection of two complex Lagrangians. Hence, it acquires a $(-1)$-shifted symplectic structure in the sense of [47]; this is the same as the shifted symplectic structure on the derived moduli stack of $\mathrm{SL}(2,\mathbb{C})$-local systems over M constructed in [47]. By [9], the fixed point locus with its shifted symplectic structure is locally the (derived) critical locus of a function and one can define Floer homology invariants of M by using the vanishing cycles of this function, see [1]. There are similar constructions in Donaldson-Thomas theory (see, for example, [4]). Such a construction can also be given in our set-up by using the potentials $V_{\sigma }$, see §6.4. Passing to the realm of wild speculation, one might ponder the possibility of similar, Floer-type, constructions on spaces of representations of the Galois group of a number field or of a local p-adic field. We say nothing more about this here. We will, however, mention that the idea of viewing certain spaces of Galois representations as Lagrangian intersections was first explained by M. Kim in [37, Sect. 10].

Classically, the Chern-Simons invariant and the volume are hard to calculate directly for closed manifolds. They can also be defined for manifolds with boundary; combined with various “surgery formulas” this greatly facilitates calculations. We currently lack examples of such calculations in our arithmetic set-up. We hope that extending the theory to non-proper curves will lead to some explicit calculations and a better understanding of the invariants. Indeed, there should be such an extension, under some assumptions. For example, we expect that there is a symplectic structure on the space of formal deformations of a representation of the fundamental group of a non-proper curve when the monodromy at the punctures is fixed up to conjugacy. We also expect that, in the case that k is an ℓ-adic field, the invariants $\mathrm{Vol}(\mathcal{F})$ and $V_{\sigma }$ can be calculated using methods of ℓ-adic Hodge theory. We hope to return to some of these topics in another paper.

Acknowledgments: We thank J. Antonio, H. Esnault, E. Kalfagianni, M. Kim, D. Litt and P. Sarnak, for useful discussions, comments and corrections. In particular, the idea of constructing a Hamiltonian vector field from the Galois action on the deformation space arose in conversations of the author with M. Kim. We also thank the IAS for support and the referee for constructive comments that improved the presentation.

Notations: Throughout the paper $\mathbb{N}$ denotes the non-negative integers and ℓ is a prime. We denote by ${\mathbb{F}}_{\ell }$ the finite field of ℓ elements and by ${\mathbb{Z}}_{\ell }$, resp. ${\mathbb{Q}}_{\ell }$, the ℓ-adic integers, resp. ℓ-adic numbers. We fix an algebraic closure ${\overline{\mathbb{Q}}}_{\ell }$ of ${\mathbb{Q}}_{\ell }$ and we denote by $|{|}_{\ell }$ (or simply $||$), resp. $v_{\ell }$, the ℓ-adic absolute value, resp. ℓ-adic valuation on ${\overline{\mathbb{Q}}}_{\ell }$, normalized so that $|\ell {|}_{\ell }={\ell }^{-1}$, $v_{\ell }(\ell )=1$. We will denote by $\mathbb{F}$ a field of characteristic ℓ which is algebraic over the prime field ${\mathbb{F}}_{\ell }$ and by $W(\mathbb{F})$, or simply W, the ring of Witt vectors with coefficients in $\mathbb{F}$. If k is a field of characteristic ≠ℓ we fix an algebraic closure $\overline{k}$. We denote by $k^{\mathrm{sep}}$ the separable closure of k in $\overline{k}$, by $k({\zeta }_{{\ell }^{\infty }})={\cup }_{n\ge 1}k({\zeta }_{{\ell }^n})$ the subfield of $k^{\mathrm{sep}}$ generated over k by the (primitive) ${\ell }^n$-th roots of unity ${\zeta }_{{\ell }^n}$ and by ${\chi }_{\mathrm{cycl}}:\mathrm{Gal}(k^{\mathrm{sep}}/k)\to {\mathbb{Z}}_{\ell }^{\times }$ the cyclotomic character defined by

$$\sigma ({\zeta }_{{\ell }^n})={\zeta }_{{\ell }^n}^{{\chi }_{\mathrm{cycl}}(\sigma )}$$

for all $n\ge 1$. We set $G_k=\mathrm{Gal}(k^{\mathrm{sep}}/k)$. Finally, we will denote by ${()}^{⁎}$ the Pontryagin dual, by ${()}^{\vee }$ the linear dual, and by ${()}^{\times }$ the units.