Bounds for twisted symmetric square L-functions via half-integral weight periods

We establish the first moment bound (phi)Sigma L(phi circle times phi Psi, 1/2) << (epsilon) p(5/4+epsilon) for triple product L-functions, where Psi is a fixed Hecke-Maass form on SL2 (Z) and phi runs over the Hecke-Maass newforms on Gamma(0) (p) of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent 5/4 is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases. Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime p, all but very few Hecke-Maass newforms on Gamma(0)(p)\H of bounded eigenvalue have very uniformly distributed mass after pushforward to SL2(Z)\H. Our main result turns out to be closely related to estimates such as (vertical bar n vertical bar p)Sigma L(Psi circle times chi(np), 1/2) << p where the sum is over those n for which np is a fundamental discriminant and chi(np) denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke-Iwaniec.ISSN:2050-509