Abstract
We derive a Motohashi-type formula for the cubic moment of central values of \(L\)-functions of level \(q\) cusp forms twisted by quadratic characters of conductor \(q\), previously studied by Conrey and Iwaniec and Young. Corollaries of this formula include Weyl-subconvex bounds for \(L\)-functions of weight two cusp forms twisted by quadratic characters, and estimates towards the Ramanujan–Petersson conjecture for Fourier coefficients of weight 3/2 cusp forms.
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Acknowledgments
I would like to express my deep appreciation to Philippe Michel, Paul Nelson, Kannan Soundararajan, Akshay Venkatesh and Matthew Young for many helpful discussions, and to thank the École Polytechnique Fédérale de Lausanne and the Swiss National Science Foundation for their generous financial support.
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The author is partially supported by Swiss National Science Foundation Grant 200021_137488.
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Petrow, I.N. A twisted Motohashi formula and Weyl-subconvexity for \(L\)-functions of weight two cusp forms. Math. Ann. 363, 175–216 (2015). https://doi.org/10.1007/s00208-014-1166-8
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DOI: https://doi.org/10.1007/s00208-014-1166-8