MOMENTS AND HYBRID SUBCONVEXITY FOR SYMMETRIC-SQUARE L-FUNCTIONS

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 vertical bar t(j)vertical bar 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 vertical bar t(j)vertical bar(6/7+delta) <= (2-delta) &VERBARt(j)&VERBAR for any fixed delta > 0. Since &VERBARt&VERBAR can be taken significantly smaller than &VERBARt(j)&VERBAR, 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.