Abstract
Fix an integer \(\kappa \geqslant 2\). Let P be prime and let \(k> \kappa \) be an even integer. For f a holomorphic cusp form of weight k and full level and g a primitive holomorphic cusp form of weight \(2 \kappa \) and level P, we prove hybrid subconvexity bounds for \(L \left( \frac{1}{2}, \text {Sym}^2 f \otimes g\right) \) in the k and P aspects when \(P^{\frac{13}{64} + \delta } < k < P^{\frac{3}{8} - \delta }\) for any \(0 < \delta < \frac{11}{128}\). These bounds are achieved through a first moment method (with amplification when \(P^{\frac{13}{64}} < k \leqslant P^{\frac{4}{13}}\)).
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Acknowledgments
We thank IAS Princeton for the wonderful working conditions in which discussions about the subconvexity problem for Rankin–Selberg L-functions were initiated between the first two authors. We thank the organizers, Valentin Blomer, Philippe Michel and Samuel Patterson of the MF Oberwolfach workshop “The Analytic Theory of Automorphic Forms”, for stimulating lectures. One lecture by Rizwanur Khan, in particular, in which ideas behind [12] were presented, triggered our interest in the specific case of \(L \left( \frac{1}{2}, \text {Sym}^2 f \otimes g\right) \). The first author completed this work through the support of the Sloan fellowship BR2011-083 and the NSF Grant DMS-1068043. The second author was partly supported by Swarna Jayanti Fellowship, 2011–2012, DST, Govt. of India. The third author is grateful to his advisor, Roman Holowinsky, who brought him to this area of research, gave him much enlightenment, guidance and encouragement.
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Holowinsky, R., Munshi, R. & Qi, Z. Hybrid subconvexity bounds for \(L \left( \frac{1}{2}, \hbox {Sym}^2 f \otimes g\right) \) . Math. Z. 283, 555–579 (2016). https://doi.org/10.1007/s00209-015-1610-9
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DOI: https://doi.org/10.1007/s00209-015-1610-9