Rational structures on automorphic representations

Abstract

This paper proves the existence of global rational structures on spaces of cusp forms of general reductive groups. We identify cases where the constructed rational structures are optimal, which includes the case of GL(n). As an application, we deduce the existence of a natural set of periods attached to cuspidal automorphic representations of GL(n). This has consequences for the arithmetic of special values of L-functions that we discuss in [30,31].In the course of proving our results, we lay the foundations for a general theory of Harish-Chandra modules over arbitrary fields of characteristic 0. In this context, a rational character theory, translation functors and an equivariant theory of cohomological induction are developed. We also study descent problems for Harish-Chandra modules in quadratic extensions, where we obtain a complete theory over number fields.

On period relations for automorphic 𝐿-functions I

“…In this section we first recall the notion of pair (a, B) over an arbitrary base field F of characteristic 0 and also the fundamental properties of the category of (a, B)-modules and (g, K)-modules over F discussed in [18]. Then we go on to define Hecke algebras for pairs (a, B) over arbitrary base fields of characteristic 0.…”

“…Then (g E , K E ) is a reductive pair over E in the sense of [18]. For the sake of readability we introduce the notation (g, K) E := (g E , K E ), that we also apply to more general pairs.…”

“…In [19] the author used the rational structures constructed in [18] to prove period relations for Rankin-Selberg L-functions. Here we generalize the method to standard L-functions of GL(2n) for cuspidal representations which are lifts from globally generic cuspidal representations of GSpin(2n + 1).…”

“…The proof of this theorem leads us to the consideration of Hard Duality for Zuckerman and Bernstein functors over arbitrary fields of characteristic 0. Zuckerman functors over arbitrary fields were already introduced by the author in [18], and their dg analogues over arbitrary rings were introduced by Hayashi [15], who also defined dg analogues of Bernstein functors over C. However Bernstein functors have not yet been defined over fields other than the complex numbers. In oder to give a unified treatament of all functors and also of the construction of functionals we generalize the appropriate Hecke algebras to arbitrary base fields of characteristic 0.…”

Beilinson–Bernstein Localization Over ℚ and Periods of Automorphic Forms: Erratum

“…If the Hom space were infinite-dimensional this would be false; however, since it is (at most) one-dimensional in the situation of Corollary 5.10, the result follows from Proposition 1.1 (iii) of [2].…”