Topic
Local zeta-function
About: Local zeta-function is a research topic. Over the lifetime, 101 publications have been published within this topic receiving 2477 citations.
Papers published on a yearly basis
Papers
Journal Article•
TL;DR: In this article, motivic analogues of Igusa's local zeta functions are defined, which take their values in a Grothendieck group of Chow motives and specialize to p-adic Igusa Local Zeta functions.
Abstract: We define motivic analogues of Igusa's local zeta functions. These functions take their values in a Grothendieck group of Chow motives. They specialize to p-adic Igusa local zeta functions and to the topological zeta functions we introduced several years ago. We study their basic properties, such as functional equations, and their relation with motivic nearby cycles. In particular the Hodge spectrum of a singular point of a function may be recovered from the Hodge realization of these zeta functions.
349 citations
TL;DR: In this article, a very explicit formula for Igusa's local zeta function Zf(s) associated to a polynomial f in several variables over the p-adic numbers, when f is sufficiently non-degenerated with respect to its Newton polyhedron Γ(f).
96 citations
Posted Content•
TL;DR: The best upper bounds on the number of rational points on an algebraic curve of genus g defined over a finite field Fq come either from Serre's refinement of the Weil bound, or from Oesterle's optimization of the explicit formulae method if the genus is large as discussed by the authors.
Abstract: Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field Fq come either from Serre's refinement of the Weil bound if the genus is small compared to q, or from Oesterle's optimization of the explicit formulae method if the genus is large. This paper presents three methods for improving these bounds. The arguments used are the indecomposability of the theta divisor of a curve, Galois descent, and Honda-Tate theory. Examples of improvements on the bounds include lowering them for a wide range of small genus when q = 2 3 ,2 5 ,2 13 ,3 3 ,3 5 ,5 3 ,5 7 , and when q = 2 2s , s > 1. For large genera, isolated improvements are obtained for q = 3,8,9.
87 citations
TL;DR: In this article, the authors provided a geometric description of the poles of the Igusa local zeta function Z(s, f) associated to an analytic mapping f = (f1,..., fl) : U(� K n )! K l, and a locally constant function Φ, with support in U, in terms of a log-principalizaton of the K (x) ideal If = (F1,,,.,,, fl).
Abstract: In this paper we provide a geometric description of the possi- ble poles of the Igusa local zeta function Z�(s, f) associated to an analytic mapping f = (f1, . . . , fl) : U(� K n ) ! K l , and a locally constant function Φ, with support in U, in terms of a log-principalizaton of the K (x) ideal If = (f1, . . . , fl). Typically our new method provides a much shorter list of possible poles compared with the previous methods. We determine the largest real part of the poles of the Igusa zeta function, and then as a corollary, we ob- tain an asymptotic estimation for the number of solutions of an arbitrary sys- tem of polynomial congruences in terms of the log-canonical threshold of the subscheme given by If . We associate to an analytic mapping f = (f1, . . . , fl) a Newton polyhedron Γ (f) and a new notion of non-degeneracy with respect to Γ (f). The novelty of this notion resides in the fact that it depends on one Newton polyhedron, and Khovanskii's non-degeneracy notion depends on the Newton polyhedra of f1, . . . , fl . By constructing a log-principalization, we give an explicit list for the possible poles of Z�(s, f), l � 1, in the case in which f is non-degenerate with respect to Γ (f).
78 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 5 |
| 2020 | 4 |
| 2019 | 4 |
| 2018 | 8 |
| 2017 | 4 |
| 2016 | 5 |