On the zeta functions of supersingular curves

A ug 2 00 4 Zeta functions of supersingular curves of genus 2

This paper was motivated by the problem of determining what isogeny classes of abelian surfaces over a finite field k contain jacobians. In [MN] we performed a numerical exploration of this problem, that led to several conjectures. We present in this paper a complete answer for supersingular surfaces in characteristic 2 (section 5). We deal with this problem in a direct way by computing explici...

Curves , Jacobians , and Zeta Functions

When we speak about a function field of one variable over a field K, we mean a finitely generated regular extension F of K of transcendence degree 1. We briefly recall the definitions of the main objects attached to F/K and their properties. See the books [Che51] or [Sti93] for details. A more comprehensive survey can be found [FrJ08, Sections 3.1-3.2]. A K-place of F is a place φ: F → K̃ ∪ {∞} ...

Computing Zeta Functions of Nondegenerate Curves

In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since in practice all known cases, e.g. hyperelliptic, superelliptic and Cab curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being no...

Holomorphy of Local Zeta Functions for Curves

Non-Abelian Zeta Functions for Elliptic Curves

In this paper, new local and global non-abelian zeta functions for elliptic curves are defined using moduli spaces of semi-stable bundles. To understand them, we also introduce and study certain refined Brill-Noether locus in the moduli spaces. Examples of these new zeta functions and a justification of using only semi-stable bundles are given too. We end this paper with an appendix on the so-c...