Abstract

Spectral reciprocities are equalities between moments of automorphic $L$-functions in different families. They are powerful tools for the study of the moment problem and the subconvexity problem. The first spectral reciprocity formula is Motohashi's formula, which relates the cubic moment of $L$-functions for $\GL_2$ with the fourth moment of $L$-functions for $\GL_1$. The exploitation of this formula (over $\mathbb{Q}$) has led Conrey-Iwaniec and Petrow-Young to the uniform Weyl bound for all Dirichlet $L$-functions. In this talk, we will briefly present a distributional version of this formula. Based on this version, applications to the uniform Weyl-type subconvex bounds for some $PGL(2)$ $L$-functions, as well as the limitation of the method, will be discussed. These results are joint works with Balkanova-Frolenkov and Ping Xi, respectively. If time permits, we will present a further application to the partition function, joint work with Nicholas Andersen. This is the first of two related talks. The other one, which deal with the distributional Motohashi's formula in detail, will be given at the department of mathematics.