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
$s=1/2$
.
Asymptotic formulae for the second moments of $ L$-series of holomorphic cusp forms on the critical line
