On the arithmetic of Shalika models and the critical values of $L$-functions for $\mathrm{GL}(2n)$

Abstract

Let $\Pi$ be a cohomological cuspidal automorphic representation of ${\mathrm{GL}}_{2n}(\mathbb{A})$ over a totally real number field $F$. Suppose that $\Pi$ has a Shalika model. We define a rational structure on the Shalika model of ${\Pi }_f$. Comparing it with a rational structure on a realization of ${\Pi }_f$ in cuspidal cohomology in top-degree, we define certain periods ${\omega }^{\epsilon }({\Pi }_f)$. We describe the behaviour of such top-degree periods upon twisting $\Pi$ by algebraic Hecke characters $\chi$ of $F$. Then we prove an algebraicity result for all the critical values of the standard $L$-functions $L(s,\Pi \otimes \chi )$; here we use the work of B. Sun on the non-vanishing of a certain quantity attached to ${\Pi }_{\infty }$. As an application, we obtain new 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 Rankin-Selberg $L$-functions for ${\mathrm{GL}}_3\times {\mathrm{GL}}_2$; assuming Langlands Functoriality, this generalizes to Rankin-Selberg $L$-functions of ${\mathrm{GL}}_n\times {\mathrm{GL}}_{n-1}$. Thirdly, for the degree four $L$-functions for ${\mathrm{GSp}}_4$. 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.

We added a few details in 3.8.1 and 3.9, where we felt, that the reader may have missed them so far. No result has changed

References in corpus (9)

• Notes on the arithmetic of Hilbert modular forms
• Transfer of Siegel cusp forms of degree 2
• Cohomologically induced distinguished representations and cohomological test vectors
• A Cuspidality Criterion for the Exterior Square Transfer of Cusp Forms on GL(4)
• Arithmeticity for periods of automorphic forms
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Cited by in corpus (3)

• Transfer of Siegel cusp forms of degree 2
• Cohomologically induced distinguished representations and cohomological test vectors
• The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions