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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)

MSC:

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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