p-Adic and analytic properties of period integrals and values of L-functions

Abstract

The Ichino-Ikeda conjecture is an identity that relates a ratio of special values of automorphic L-functions to a ratio of period integrals. Both sides of this identity are expected to satisfy certain equidistribution properties when the data vary, and indeed it has been possible to transfer such properties from one side of the identity to the other in cases where the identity is known. The present article studies parallels between complexanalytic and p-adic equidistribution properties and relates the latter to questions about Galois cohomology. Keywords Galois representation · Shimura variety · Special values of L-functionsRésumé La conjecture d'Ichino-Ikeda est une identité qui met en relation un rapport entre valeurs spéciales de fonctions L automorphes et un rapport entre périodes d'intégrales. On attend à ce que les deux côtés de cette identité vérifient certaines propriétés d'équidistribution lorsqu'on fait varier les données ; en effet, dans certains cas on a réussi à transférer de telles propriétés d'un côté de l'identité à l'autre. Cet article développe des parallèles entre les propriétés d'équidistribution dans le cadre complexe-analytique et dans le cadre p-adique; dans la situation p-adique ces propriétés sont liées à des questions sur la cohomologie galoisienne.

On critical values of L-functions of potentially automorphic motives

“…Finally, since the hermitian space V is totally definite, the quadratic period Q hol (π ) can be taken to be in E(π ) (see for instance [20], Section 5). Thus, it can taken to be 1 in (3.3.2), which, together with (3.3.3), proves the formula in the statement of the theorem.…”

On the non-vanishing of the central value of certain $L$-functions: unitary groups

“…For higher rank unitary groups, the non-vanishing of the central value L( 1 2 , π × χ) has been formulated as a basic input towards the investigation of many interesting and important problems related to the theory of motives, arithmetic geometry, and the theory of p-adic L-functions. We refer to some recent papers of M. Harris ([7] and [8]) for more detailed account of the significance of such non-vanishing property. The objective of this paper is to understand such a non-vanishing problem in a framework containing:…”