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LEVEL RECIPROCITY IN THE TWISTED SECOND MOMENT OF RANKIN–SELBERG $L$-FUNCTIONS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 June 2018

Nickolas Andersen  and
Eren Mehmet Kıral
Nickolas Andersen
Affiliation:
UCLA Mathematics Department, Los Angeles, CA 90095, U.S.A. email nandersen@math.ucla.edu
Eren Mehmet Kıral
Affiliation:
Wako-Shi, Saitama, Japan email erenmehmetkiral@protonmail.com
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Abstract

We prove an exact formula for the second moment of Rankin–Selberg $L$-functions $L(\frac{1}{2},f\times g)$ twisted by $\unicode[STIX]{x1D706}_{f}(p)$, where $g$ is a fixed holomorphic cusp form and $f$ is summed over automorphic forms of a given level $q$. The formula is a reciprocity relation that exchanges the twist parameter $p$ and the level $q$. The method involves the Bruggeman–Kuznetsov trace formula on both ends; finally the reciprocity relation is established by an identity of sums of Kloosterman sums.

MSC classification

Primary: 11F03: Modular and automorphic functions 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11F72: Spectral theory; Selberg trace formula
Type
Research Article
Information
Mathematika , Volume 64 , Issue 3 , 2018 , pp. 770 - 784
Copyright
Copyright © University College London 2018 

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Footnotes

This material is based upon work supported by the National Science Foundation under Grant No. 1440140, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring semester of 2017. The first author is also supported by NSF grant DMS-1701638.

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