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 is a sum over $t_j$ in short interval. At central point $s=1/2$ $L$-function, our interval smaller than previous known results. More specifically, $|t_j|$ size $T$, $T^{1/5}$, while best was $T^{1/3}$ from work Lam. A little higher up on critical line, yields subconvexity bound $L$-function. we get at $s=1/2+it$ provided $|t_j|^{6/7+\delta}\le |t| \le (2-\delta)|t_j|$ any fixed $\delta>0$. Since $|t|$ can be taken significantly $|t_j|$, this may viewed as an approximation notorious problem $L$-function aspect $s=1/2$.