## 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.

#### Files

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