Barcelona Fall Workshop on Number Theory II

Integral presentations of $GL(n)\ast GL(2)$ Rankin-Selberg L-functions and applications

Presenters

Jeanine van Order (Bielefeld)

Abstract

Central and critical values of Rankin-Selberg L-functions for $GL(n)\ast GL(2)$ (for $n \geq 2$) play a major underlying role in arithmetic geometry, starring in the conjectures Birch-Swinnerton-Dyer, Iwasawa-Greenberg, and Deligne, not to mention various open conjectures in the analytic theory of automorphic forms. I would like to explain how several features of the underlying representation theory, particularly the surjectivity of the archimedean local Kirillov map and a certain classical projection operator (used e.g. to establish converse theorems) lead to novel integral presentations of these values as the constant coefficients of certain $L^2$-automorphic forms on the mirabolic subgroup of $GL(2)$. Making a suitable extension to $GL(2)$ then gives a convenient re-interpretation of such values, for instance to study central critical values in families. In this latter setting, I will explain a novel approach to deriving nonvanishing estimates via spectral decompositions of Eisenstein series, as well as the relation to recent progress on Deligne’s conjecture for automorphic motives, and (if time permits) the potential to give new constructions of p-adic interpolation series.