On the arithmetic of Shalika models and the critical values of L -functions for GL 2 n

Abstract

Abstract. Let Π be a cohomological cuspidal automorphic representation of GL 2n (A) over a totally real number field F . Suppose that Π has a Shalika model. We define a rational structure on the Shalika model of Π f . Comparing it with a rational structure on a realization of Π f in cuspidal cohomology in top-degree, we define certain periods ω ǫ (Π f ). We describe the behaviour of such top-degree periods upon twisting Π by algebraic Hecke characters χ of F . Then we prove an algebraicity result for all the critical values of the standard L-functions L(s, Π ⊗ χ); here we use the recent work of B. Sun on the non-vanishing of a certain quantity attached to Π∞. As applications, we obtain algebraicity results in the following cases: Firstly, for the symmetric cube L-functions attached to holomorphic Hilbert modular cusp forms; we also discuss the situation for higher symmetric powers. Secondly, for certain (self-dual of symplectic type) Rankin-Selberg L-functions for GL 3 × GL 2 ; assuming Langlands Functoriality, this generalizes to certain Rankin-Selberg L-functions of GLn × GL n−1 . Thirdly, for the degree four L-functions attached to Siegel modular forms of genus 2 and full level. Moreover, we compare our top-degree periods with periods defined by other authors. We also show that our main theorem is compatible with conjectures of Deligne and Gross.

On period relations for automorphic 𝐿-functions I

“…We remark that if Π is cohomological with associated weight w 1 as in (40), then (cf. Lemma 3.6.1 in [12]),…”

“…In this section we recall the theory of Friedberg-Jacquet [10,16] in the context of Shalika models for GL(2n). For a nice exposition of the subject, we refer to Section 3 in [12].…”

“…This implies that η is an algebraic Hecke character and therefore Q(η) is a number field. Furthermore by Wee Teck Gan's Theorem in the Appendix of [12], if Π admits an (η, ψ)-Shalika model, then for each σ ∈ Aut(C/Q) the twisted representation Π σ admits an (η σ , ψ)-Shalika model.…”

“…The period Ω(χ ∞ ) ∈ C × in the global special value formula Λ( 1 2 , Π ⊗ χ) G(χ) m · Ω(χ ∞ ) ∈ Q(µ, χ) (cf. Theorem 7.2.1 in [12]), arises as follows. We fix for each signature δ ∈ {0, 1} rF a generator…”

Uniqueness of twisted linear periods and twisted Shalika periods

“…Using the Langlands lift to GL 2n , local Shalika periods and their global analogues are fundamental to the study of standard L-functions of GSpin 2n+1 . See [GR,Section 3] or [AsG] for example. Similar to the untwisted case [JR, AGJ], the proof of Theorem A is based on Shalika zeta integrals [FJ] and the following uniqueness result.…”