# Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)

Mathematische Annalen, Sep 2017

Let $G=\mathrm{GL}_{2n}$ over a totally real number field F and $n\ge 2$. 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 paper, we prove 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 Gauß sums of Hecke characters.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00208-017-1590-7.pdf

Harald Grobner. Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe), Mathematische Annalen, 2017, 1-41, DOI: 10.1007/s00208-017-1590-7