Séminaire de Mathématique

# Rationality results for the symmetric and exterior square L-function of GL(2n)

## par Prof. Harald GROBNER (Université de Vienne)

Europe/Paris
Amphithéâtre Léon Motchane (IHES)

### Amphithéâtre Léon Motchane

#### IHES

Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette
Description

ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

Let $G$ be GL$(2n)$ over a totally real number field $F$, $n\geq 2$. Let $\Pi$ be a cuspidal automorphic representation of $G(\mathbb A)$, which is cohomological and a functorial lift from SO$(2n+1)$. The latter condition can be equivalently reformulated that the exterior square $L$-function of $\Pi$ has a pole at $s=1$. In this talk, we present a rationality result for the residue of the exterior square $L$-function at $s=1$ and also for the holomorphic value of the symmetric square $L$-function at $s=1$ attached to $\Pi$. As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square $L$-functions and a product of Gau\ss ~sums of Hecke characters.

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