A trace formula for the scalar product of Hecke series and its applications

Abstract

A trace formula expressing the mean values of the form (k = 2, 3,... ) ! via certain arithmetic means on the group F0(N1 ) is proved. Here the sum is taken over a normalized orthogonal basis in the space of holomorphic cusp forms of weight 2k with respect to F0(N1).

Subconvexity for twisted L-functions over number fields via shifted convolution sums

“…The family of twisted L-functions considered in Theorem 1 was the first instance of the automorphic subconvexity problem to be studied systematically (see for example the works [Iwa87], [Duk88], [DFI93], [Byk96], [CI00], [CPSS], [BHM07], [Ven10]). Via Waldspurger type formulae, critical values of twisted L-functions are connected to Fourier coefficients of modular forms of half-integral weight.…”

A Burgess-like subconvex bound for twisted L-functions

“…If g is holomorphic of level D = 1 they proved the subconvex exponent (1.2) 1 2 − 1 22 , using the δ-symbol method. In the case of a general holomorphic cusp form of weight at least 2, Bykovskiȋ [By96] derived, by a different method, the stronger subconvex exponent (1.3) 1 2 − 1 8 as long as (D, q) = 1. While it is unclear whether and to what extent Bykovskiȋ's method carries over to the general case (1.1), the second and third author independently used the strategy from [DFI93] to break convexity also in the Maass case [H03a, H03b, M04a].…”

Addendum: Hybrid bounds for twisted L-functions

“…It equals Here we see that the display after [3, (1.3)] has an additional factor .a 2 /, and hence the second display after [3, (1.4)] has an additional factor (3) . m 2 C c 2 q=N /;…”

“…in the notation of [3]. In the notation of [1], q=N in (3) equals c=q, which in turn is divisible by D=q.…”