Periods and nonvanishing of central L-values for GL(2n)
Brooke Feigon, Kimball Martin, David Whitehouse
TL;DR
This work proves Guo–Jacquet’s conjecture on the relation between the nonvanishing of central L-values for GL(2n) and GL_n(E) periods by employing a simple relative trace formula and local matching of orbital integrals. The authors leverage Ramakrishnan's mult1 to bypass the full fundamental lemma and obtain global-to-local statements: if a cuspidal π on G is H-distinguished, its Jacquet–Langlands transfer π′ on G′ is symplectic with L(1/2, π′_E) ≠ 0; they also derive converse results under certain local hypotheses and establish global transfer phenomena across quaternionic forms for even n. The paper then translates these global results into local consequences for distinguished supercuspidal representations, providing partial confirmations of PTB and FM conjectures in the local setting. The methods blend a carefully constructed relative trace formula with local orbital-integral matching and a detailed analysis of local Bessel distributions, yielding a higher-rank generalization of Waldspurger-type phenomena and illuminating the local-global interplay for distinguished representations and central L-values.
Abstract
Let $π$ be a cuspidal automorphic representation of PGL($2n$) over a number field $F$, and $η$ the quadratic idele class character attached to a quadratic extension $E/F$. Guo and Jacquet conjectured a relation between the nonvanishing of $L(1/2,π)L(1/2, π\otimes η)$ for $π$ 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.