Periods and nonvanishing of central L-values for GL(2n)

Abstract

Let $\pi$ be a cuspidal automorphic representation of PGL($2n$) over a number field $F$, and $\eta$ the quadratic idele class character attached to a quadratic extension $E/F$. Guo and Jacquet conjectured a relation between the nonvanishing of $L(1/2,\pi )L(1/2,\pi \otimes \eta )$ for $\pi$ of symplectic type and the nonvanishing of certain GL($n,E$) periods. When $n=1$, this specializes to a well-known result of Waldspurger. We prove this conjecture, and related global results, under some local hypotheses using a simple relative trace formula. We then apply these global results to obtain local results on distinguished supercuspidal representations, which partially establish a conjecture of Prasad and Takloo-Bighash.

31 pages; to appear in Israel J. Math

References in corpus (6)

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• Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations
• Automorphic period and the central value of Rankin--Selberg L-function
• On the smooth transfer for Guo-Jacquet relative trace formulae
• An archimedean analog of Jacquet - Rallis theorem
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Cited by in corpus (9)

• On the smooth transfer for Guo-Jacquet relative trace formulae
• An infinitesimal variant of Guo-Jacquet trace formula II
• An infinitesimal variant of Guo-Jacquet trace formula I: the case of $(G{L}_{2n,D},G{L}_{n,D}\times G{L}_{n,D})$
• Local periods for discrete series representations
• A note on local periods for supercuspidal representations
• Intersections in Lubin-Tate space and biquadratic fundamental lemmas
• On the fine expansion of the unipotent contribution of the Guo-Jacquet trace formula
• A local trace formula for $p$-adic infinitesimal symmetric spaces: the case of Guo-Jacquet
• Strong local-global phenomena for Galois and automorphic representations