On Certain Period Relations for Cusp Forms on GLn

Abstract

Let π be a regular algebraic cuspidal automorphic representation of GL n (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 = 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.

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

“…This us essentially due to the uniqueness of essential vectors for Π v , v / ∈ S ∞ : Otherwise put, the proof of Prop./Def. 3.3 in Raghuram-Shahidi [29] goes through word for word in our (slightly different) situation at hand. The (Whittaker) periods p t (Π) defined by Prop.…”

“…This is the analogue of Prop. 4.2, whose proof goes through word for word in the current situation, i.e., for cohomology in degree b instead of t. See [29], Prop./Def. 3.3.…”

“…4.2.1. The idea behind the construction of p t (Π) (as of ω ǫ (Π f )), however, goes back to [14], [25] and [29].…”

Eisenstein cohomology and ratios of critical values of Rankin–Selberg L-functions

“…where each piece on the right hand side is one-dimensional, and ǫ is a canonical sign (see [10,Section 3.3]). …”

“…Since cuspidal automorphic representations of GL n are globally generic we can also define periods by comparing rational structures on Whittaker models and cohomological realizations. The periods were denoted p ± (π f ) in Raghuram-Shahidi [10] and they appear in algebraicity results for the central critical value of Rankin-Selberg L-functions for GL n × GL n−1 ; see Raghuram [9, Theorem 1.1]. The periods p ± (π f ) depend on a choice of a nontrivial character of Q\A which is implicit in any discussion concerning Whittaker models.…”