The subconvexity problem for GL2

Abstract

Generalizing and unifying prior results, we solve the subconvexity problem for the L-functions of GL 1 and GL 2 automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino-Ikeda.

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

“…We remark that the same exponent was simultaneously achieved by Wu [Wu14], using a method built on [MV10]. 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]).…”

“…Dirichlet L-functions), the famous Burgess bound [Bur63] is the above with δ = 3/16. For automorphic GL 2 L-functions over number fields, the subconvexity problem was solved by Michel and Venkatesh [MV10] with an unspecified δ . Recently, Blomer and Harcos [BH10] proved a Burgess type subconvex bound for twisted automorphic GL 2 L-functions over totally real number fields.…”

On signs of Fourier coefficients of cusp forms

“…This shows that the current method is incapable to show that n f Q 0.372 . Using a subconvexity bound would slightly push the limit: Michel and Venkatesh [17] have recently proved that one can replace 1/4 + in (2·1) by η for some η which is slightly smaller than 1/4. Remark 9.…”

Distribution of periodic torus orbits and Duke's theorem for cubic fields

“…For n = 3 the bound (7) follows from the work of Burgess [6] if K is abelian and (essentially) from the deep work of Duke, Friedlander and Iwaniec if K is cubic not abelian ( [3], [14], [37]). …”