On certain period relations for cusp forms on GL_n

TL;DR

This paper studies special periods attached to automorphic representations on GL_n, analyzing their behavior under twists and relating them to Deligne's conjecture for symmetric power L-functions, especially for dihedral forms.

Contribution

It extends the understanding of periods for GL_n automorphic forms, analyzes their behavior under algebraic twists, and provides new insights into Deligne's conjecture for dihedral modular forms.

Findings

Periods behave predictably under twisting by algebraic Hecke characters.

Established a relation between periods of mbda^n and mbda for dihedral forms.

Provided a new proof of Deligne's conjecture for symmetric power L-values in dihedral cases.

Abstract

Let $\pi$ be a regular algebraic cuspidal automorphic representation of ${\rm GL}_n({\mathbb A}_F)$ for a number field $F$. We consider certain periods attached to $\pi$. These periods were originally defined by Harder when $n=2$, and later by Mahnkopf when $F = {\mathbb Q}$. In the first part of the paper we analyze the behaviour of these periods upon twisting $\pi$ by algebraic Hecke characters. In the latter part of the paper we consider Shimura's periods associated to a modular form. If $\phi_{\chi}$ is the cusp form associated to a character $\chi$ of a quadratic extension, then we relate the periods of $\phi_{\chi^n}$ to those of $\phi_{\chi}$, and as a consequence give another proof of Deligne's conjecture on the critical values of symmetric power $L$-functions associated to dihedral modular forms. Finally, we make some remarks on the symmetric fourth power $L$-functions.