La m\'ethode de Shintani et ses variantes

TL;DR

This paper explores various versions of Shintani's method for decomposing fundamental domains of tori into simplicial cones and applies these techniques to study Hecke L-functions, including proving the class number formula at s=0.

Contribution

The paper introduces multiple variants of Shintani's method and demonstrates their application to key problems in number theory, such as the analytic class number formula.

Findings

Decomposition of fundamental domains into simplicial cones using Shintani's variants

Application of these methods to analyze Hecke L-functions at integer points

Proof of the analytic class number formula for totally real fields at s=0

Abstract

We give several versions of Shintani's method for the decomposition into simplicial cones of the fundamental domain of a torus modulo a lattice, and we investigate some applications to the study of Hecke $L$-functions at integer points. In particular, we prove the analytic class number formula for a totally real field, directly at $s=0$.