On Certain Period Relations for Cusp Forms on GLn

By A. Raghuram, F. Shahidi. 2007

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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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eeafbeba31e06537cbd0d87c9bc34705d46b8379

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Citations in the corpus

NOTES ON THE ARITHMETIC OF HILBERT MODULAR FORMS (2011, background): “The following result on all critical values for twisted L-functions follows from the period relations proved in [34].”
On the arithmetic of Shalika models and the critical values of L-functions for GL2n (2011, background): “Then, Πv is cohomological with respect to E v µv for all v ∈ S∞ by Raghuram–Shahidi [43, Thm. 5.5] and so, using the Künneth rule, Π is cohomological with respect to the highest weight module Evµ with µ = (µv)v∈S∞ .”
On the Special Values of Certain Rankin-Selberg L-Functions and Applications to Odd Symmetric Power L-Functions of Modular Forms (2008, methodology): “We review the very basic ingredients here, and refer the reader to [33] for all finer details.”
Critical values of Rankin-Selberg L-functions for GL(n) x GL(n-1) and the symmetric cube L-functions for GL(2) (2013, methodology): “The reader is referred to my paper with Shahidi [33] for all the details.”
WHITTAKER PERIODS, MOTIVIC PERIODS, AND SPECIAL VALUES OF TENSOR PRODUCT $L$ -FUNCTIONS (2013, methodology): “This section is in analogy with Mahnkopf [30], Raghuram-Shahidi, [34], Raghuram [33] and Grobner-Raghuram [15].”