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Register a new userZeta functions of a class of Artin-Schreier curves with many automorphisms
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This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta functions of the curves in the class over the field of definition of all automorphisms in the subgroup. As a consequence, we obtain new examples of maximal curves.
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We study the canonical model of the Artin-Schreier-Kummer-Witt flat family of curves over a ring of mixed characteristic. We first prove the relative version of a classical theorem by Petri, then use the model proposed by Bertin-Mezard to construct an explicit generating set for the relative canonical ideal. As a byproduct, we obtain a combinatorial criterion for a set to generate the canonical ideal, applicable to any curve satisfying the assumptions of Petris theorem.
Let ${cal M}_{g,[n]}$, for $2g-2+n>0$, be the D-M moduli stack of smooth curves of genus $g$ labeled by $n$ unordered distinct points. The main result of the paper is that a finite, connected etale cover ${cal M}^l$ of ${cal M}_{g,[n]}$, defined over a sub-$p$-adic field $k$, is almost anabelian in the sense conjectured by Grothendieck for curves and their moduli spaces. The precise result is the following. Let $pi_1({cal M}^l_{ol{k}})$ be the geometric algebraic fundamental group of ${cal M}^l$ and let ${Out}^*(pi_1({cal M}^l_{ol{k}}))$ be the group of its exterior automorphisms which preserve the conjugacy classes of elements corresponding to simple loops around the Deligne-Mumford boundary of ${cal M}^l$ (this is the $ast$-condition motivating the almost above). Let us denote by ${Out}^*_{G_k}(pi_1({cal M}^l_{ol{k}}))$ the subgroup consisting of elements which commute with the natural action of the absolute Galois group $G_k$ of $k$. Let us assume, moreover, that the generic point of the D-M stack ${cal M}^l$ has a trivial automorphisms group. Then, there is a natural isomorphism: $${Aut}_k({cal M}^l)cong{Out}^*_{G_k}(pi_1({cal M}^l_{ol{k}})).$$ This partially extends to moduli spaces of curves the anabelian properties proved by Mochizuki for hyperbolic curves over sub-$p$-adic fields.
Contents: Rational functions with given monodromy on generic curves (I. Bouw & S. Wewers); Can deformation rings of group representations not be local complete intersections? (T. Chinburg); Lifting an automorphism group to finite characteristic (G. Cornelissen); Flat connections and representations of the fundamental group in characteristic p>0 (C. Gasbarri) Questions on p-torsion of hyperelliptic curves (D. Glass & R. Pries); Automorphisms of curves and stable reduction (C. Lehr & M. Matignon); p-groups as automorphism groups of curves in characteristic p; Lifting Galois covers of smooth curves (M. Matignon); Abelian varieties isogenous to a Jacobian (F. Oort); Minimal maximal number of automorphisms of curves (F. Oort)
Using $ell$-adic cohomology of tensor inductions of lisse $overline{mathbb Q}_ell$-sheaves, we study a class of incomplete character sums.
We fix a monic polynomial $f(x) in mathbb F_q[x]$ over a finite field and consider the Artin-Schreier-Witt tower defined by $f(x)$; this is a tower of curves $cdots to C_m to C_{m-1} to cdots to C_0 =mathbb A^1$, with total Galois group $mathbb Z_p$. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function form arithmetic progressions which are independent of the conductor of the character. As a corollary, we obtain a result on the behavior of the slopes of the eigencurve associated to the Artin-Schreier-Witt tower, analogous to the result of Buzzard and Kilford.
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