Double first moment for [formula omitted] by applying Petersson's formula twice

Abstract

For holomorphic cusp forms f of weight k1 and g of weight k2 for SL(2,Z), bounds are proved for sums of L(12,Sym2f×g) over both f and g. Since these central values are known to be non-negative, the Lindelöf Hypothesis on average for both f and g follows. As a consequence, bounds for sums of the central values over g are proved for any f with its weight k1 tending to ∞ in certain ways. Subconvexity bounds for individual central values are also established in the two weight aspects for all f and all but a relatively small number of exceptional g. As an application, subconvexity bounds for the triple product L-function L(s,f×f×g) are also established. These subconvexity bounds for non-exceptional g's allow f to move and exceed the strength of all known bounds and the Weyl-type bound.