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arXiv · 1911.06310

Eisenstein series and the cubic moment for PGL(2)

Paul D. Nelson
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Abstract

Following a strategy suggested by Michel--Venkatesh, we study the cubic moment of automorphic $L$-functions on $\operatorname{PGL}_2$ 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.

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Paul D. Nelson. 2019-11-14. Eisenstein series and the cubic moment for PGL(2). https://arxiv.org/abs/1911.06310

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