On certain period relations for cusp forms on GL_n
Abstract
Description
Let $π$ 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 $π$. 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 $π$ by algebraic Hecke characters. In the latter part of the paper we consider Shimura's periods associated to a modular form. If $ϕ_χ$ is the cusp form associated to a character $χ$ of a quadratic extension, then we relate the periods of $ϕ_{χ^n}$ to those of $ϕ_χ$, 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.
40 pages. This preprint is also up on preprint server of the Erwin Schrodinger Institute as preprint number 1928. The URL is http://www.esi.ac.at/Preprint-shadows/esi1928.html
40 pages. This preprint is also up on preprint server of the Erwin Schrodinger Institute as preprint number 1928. The URL is http://www.esi.ac.at/Preprint-shadows/esi1928.html