Higher-rank zeta functions andSLn-zeta functions for curves

Selberg zeta functions for spaces of higher rank

5 Introduction In 1956 A. Selberg introduced the zeta function Z(s) = c N ≥0 (1 − e −(s+N)l(c)), Re(s) >> 0, where the first product is taken over all primitive closed geodesics in a compact Riemannian surface of genus ≥ 2, equipped with the hyperbolic metric, and l(c) denotes the length of the geodesic c. Selberg proved that the product converges if the real part of s is large enough and that ...

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 ∪ {∞} ...

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...

Holomorphy of Local Zeta Functions for Curves

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...