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MOMENTS AND HYBRID SUBCONVEXITY FOR SYMMETRIC-SQUARE L-FUNCTIONS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 22 / Issue 5 / September 2023
- Published online by Cambridge University Press:
- 06 December 2021, pp. 2029-2073
- Print publication:
- September 2023
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- Article
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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 
$s=1/2$.
