Eisenstein series and the cubic moment for PGL(2)

Abstract

Following a strategy suggested by Michel--Venkatesh, we study the cubic moment of automorphic $L$-functions on ${\mathrm{PGL}}_2$ using regularized diagonal periods of products of Eisenstein series. Our main innovation is to produce vectors whose integral transforms achieve arbitrarily weighted moments. Applications include general Motohashi-type identities and Weyl-type subconvex bounds for some families of $L$-functions, extending some results of Conrey--Iwaniec and Petrow--Young to the number field setting. We deduce improved estimates for representation numbers of ternary quadratic forms over number fields and for the prime geodesic theorem on arithmetic hyperbolic $3$-folds.

77 pages. minor clarifications added

References in corpus (1)

• Mean Values of Zeta-Functions via Representation Theory

Cited by in corpus (5)

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• Subconvexity for $L$-Functions on ${\mathrm{GL}}_3$ over Number Fields