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MOMENTS AND HYBRID SUBCONVEXITY FOR SYMMETRIC-SQUARE L-FUNCTIONS

Part of: Zeta and $L$-functions: analytic theory Partial differential equations on manifolds; differential operators Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  06 December 2021

Rizwanur Khan Open the ORCID record for Rizwanur Khan [Opens in a new window]  and
Matthew P. Young
Rizwanur Khan*
Affiliation:
Department of Mathematics, University of Mississippi, University, MS 38677
Matthew P. Young
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368 (myoung@math.tamu.edu)
*
rrkhan@olemiss.edu
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Abstract

We establish sharp bounds for the second moment of symmetric-square L-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$, where the second moment is a sum over $t_j$ in a short interval. At the central point $s=1/2$ of the L-function, our interval is smaller than previous known results. More specifically, for $\left \lvert t_j\right \rvert $ of size T, our interval is of size $T^{1/5}$, whereas the previous best was $T^{1/3}$, from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square L-function. More specifically, we get subconvexity at $s=1/2+it$ provided $\left \lvert t_j\right \rvert ^{6/7+\delta }\le \lvert t\rvert \le (2-\delta )\left \lvert t_j\right \rvert $ for any fixed $\delta>0$. Since $\lvert t\rvert $ can be taken significantly smaller than $\left \lvert t_j\right \rvert $, this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square L-function in the spectral aspect at $s=1/2$.

Keywords

L-functionssymmetric-squaremomentssubconvexityMaass formsquantum unique ergodicity

MSC classification

Primary: 11M41: Other Dirichlet series and zeta functions 11F12: Automorphic forms, one variable
Secondary: 58J51: Relations between spectral theory and ergodic theory, e.g. quantum unique ergodicity
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 5 , September 2023 , pp. 2029 - 2073
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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