Igusa's Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials

We study the twisted local zeta function associated to a polynomial in two variables with coefficients in a non-Archimedean local field of arbitrary characteristic. Under the hypothesis that the polynomial is arithmetically non degenerate, we obtain an explicit list of candidates for the poles in terms of geometric data obtained from a family of arithmetic Newton polygons attached to the polynomial. The notion of arithmetical non degeneracy due to Saia and Z\'u\~niga-Galindo is weaker than the usual notion of non degeneracy due to Kouchnirenko. As an application we obtain asymptotic expansions for certain exponential sums attached to these polynomials.Comment: 20 pages. In this version there is a more precise statement of Lemma 2.4 and a correction to the Example in Section 4. Minor corrections adde

Motivic Zeta Functions on \mathds{Q}-Gorenstein Varieties

We study motivic zeta functions for \mathds{Q}-divisors in a \mathds{Q}-Gorenstein variety. By using a toric partial resolution of singularities we reduce this study to the local case of two normal crossing divisors where the ambient space is an abelian quotient singularity. For the latter we provide a closed formula which is worked out directly on the quotient singular variety. As a first application we provide a family of surface singularities where the use of weighted blow-ups reduces the set of candidate poles drastically. We also present an example of a quotient singularity under the action of a nonabelian group, from which we compute some invariants of motivic nature after constructing a \mathds{Q}-resolution.Comment: 27 pages, 4 figures. New version with minor correction

Una introducción a la teoría de las funciones Zeta locales para principiantes

This survey article aims to provide an introduction to the theory of local zeta functions in the p-adic framework for beginners. We also give an extensive guide to the current literature on local zeta functions and its connections with other fields in mathematics and physics.En este artículo panorámico brindamos una introducción a la teoría de las funciones zeta locales p-ádicas para principiantes. También se presenta una revisión extensiva a la literatura especializada sobre funciones zeta locales y sus conexiones con otros campos de las matemáticas y la física

An Introduction to the Theory of Local Zeta Functions from Scratch

This survey article aims to provide an introduction to the theory of local zeta functions in the p-adic framework for beginners. We also give an extensive guide to the current literature on local zeta functions and its connections with other fields in mathematics and physics

An Introduction to the Theory of Local Zeta Functions from Scratch

En este artículo panorámico brindamos una introducción a la teoría de las funciones zeta locales p-ádicas para principiantes. También se presenta una revisión extensiva a la literatura especializada sobre funciones zeta locales y sus conexiones con otros campos de las matemáticas y la física.This survey article aims to provide an introduction to the theory of local zeta functions in the p-adic framework for beginners. We also give an extensive guide to the current literature on local zeta functions and its connections with other fields in mathematics and physics