Counting local systems with principal unipotent local monodromy. (English) Zbl 1284.14026

Let \(X_1\) be a smooth geometrically connected projective curve over the finite field of \(q\)-elements \(\mathbb{F}_q\). Let \(S_1\subseteq X_1\) be a reduced divisor consisting of \(N_1\) closed points. Set \(\mathbb{F}\) to be the algebraic closure of \(\mathbb{F}_q\). Let \((X,S):=(X_1, S_1)\otimes_{\mathbb{F}_q}\mathbb{F}\), \(l\) a prime number not dividing \(q\). There is an equivalence of categories between the category of \(\bar{\mathbb{Q}}_l\)-lisse sheaves and the category of continous finite dimensional \(\bar{\mathbb{Q}}_l\)-representations of \(\pi_1^{\text{ét}}(X-S,x)\), where \(x\in (X-S)(\mathbb{F})\) is a geometric point on which the equivalence functor depends. Since \(X-S\) is obtained from \(X_1-S_1\) by base change, the Galois group \({\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\) acts on the scheme \(X-S\) and hence on the set of isomorphism classes of \(\bar{\mathbb{Q}}_l\)-lisse sheaves, or equivalently on the set of isomorphism classes of continous finite dimensional \(\bar{\mathbb{Q}}_l\)-representations of \(\pi_1^{\text{ét}}(X-S,x)\), via transport of structures. There are several equivalent ways to describe this action. One way is to look at the fundamental exact sequence of the étale fundamental groups \[ 1\to \pi_1^{\text{ét}}(X-S,x)\to \pi_1^{\text{ét}}(X_1-S_1,x) -^{{\phi}}\rightarrow {\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\to 1. \] For any \(\sigma\in{\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\) we choose an element \(\tau\in \phi^{-1}(\sigma)\), then for any continous finite dimensional representation \(\rho: \pi_1^{\text{ét}}(X-S,x)\to \text{GL}(V)\) we have \(\sigma(\rho)=( g\mapsto \rho(\tau^{-1}g\tau))\) for all \(g\in\pi_1^{\text{ét}}(X-S,x)\). As an action of \({\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\) on the isomorphism classes of \(\bar{\mathbb{Q}}_l\)-lisse sheaves, the so defined action does not depend on the choice of the element \({\tau\in \phi^{-1}(\sigma)}\). Let \(F_1\) be the fraction field of \(X_1\), \(F=F_1\otimes_{\mathbb{F}_q}\mathbb{F}\) be the fraction field of \(X\), \(s\in S\). Then the choice of a place \(\bar{s}\) of \(\bar{F}\) above \(s\) defines an inertia group \(I_s\subset {\text{Gal}}(\bar{F}/F)\). A \(\bar{\mathbb{Q}}_l\)-lisse sheaf is said to have “Principal unipotent local monodromy at \(s\)” if the composition \(I_s\subset {\text{Gal}}(\bar{F}/F)\twoheadrightarrow \pi_1^{\text{ét}}(X-S,x)-^{\rho}\rightarrow {\text{GL}}(V)\) factors through the largest pro-\(l\) quotient (\(\cong \mathbb{Z}_l\)) of \(I_s\) with an element of \(I_s\) with image \(a\) in \( \mathbb{Z}_l\) acting on \(V\) as exp\((aN)\), where \(N\) is nilpotent with one Jordan block. Let \(\mathcal{T}^{(n)}(X,S)\) be the set of isomorphism classes of rank \(n\) irreducible \(\bar{\mathbb{Q}}_l\)-smooth sheaves on \(X-S\), with principal unipotent local monodromy at each \(s\in S\). \(\mathcal{T}^{(n)}(X,S)\) as a subset of the isomorphism classes of \(\bar{\mathbb{Q}}_l\)-lisse sheaves is stable under the \({\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\)-action. Let \(T(X_1,S_1,n)\) denote the number of fixed points of \(\mathcal{T}^{(n)}(X,S)\) by the geometric Frobenius \(\text{Frob}\in{\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\). For each \(m\geq 1\), let \((X_m,S_m):=(X_1,S_1)\otimes_{\mathbb{F}_q}\mathbb{F}_{q^m}\), then \(T(X_1,S_1,n,m):= T(X_m,S_m,n)\), where \((X_m,S_m)\) is viewed as a pair over \(\mathbb{F}_{q^m}\).
The aim of the article under review is to give a computation of the number \(T(X_1,S_1,n,m)\). It starts with a formula for \(T(X_1,S_1,n)\), under the assumption that \(n\) and \(N_1\) are \(\geq 2\), in terms of \(N_1\), \(n\), \(q\), the degrees \(\deg(s)\) for \(s\in S_1\) and the coefficients of the polynomial \(f(t):= \det(1-\text{Frob}\cdot t, H^1(X))\), where \({\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\) acts on the \(l\)-adic cohomology group \(H^1(X)\) by transport of structures. However, the first formula is not helpful to understand how the number of fixed points varies with \(m\). One problem is that when one replaces \((X_1,S_1)/\mathbb{F}_q\) by \((X_m,S_m)/\mathbb{F}_{q^m}\), the number \((n/S_1):=\{\text{the largest divisor of }n\) that is prime to all \(\deg(s)\) for \(s\in S_1\)
Reviewer: Lei Zhang (Berlin)


14F20 Étale and other Grothendieck topologies and (co)homologies
11F75 Cohomology of arithmetic groups
14G15 Finite ground fields in algebraic geometry
14G35 Modular and Shimura varieties
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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