Description
Abstract: Asymptotic evaluation of higher moments of higher degree $L$-values is an interesting problem and has potential applications towards many questions in analytic theory of automorphic forms, e.g. subconvexity of the central $L$-values. In this talk I will explain a recent result on asymptotic evaluation of the second moment of $\mathrm{GL}(n) \times\mathrm{GL}(n)$ Rankin-Selberg central $L$-values where one of the forms is a fixed cuspidal representation and the other form is varying in a family containing representations with analytic conductors bounded by $X$ and $X \to \infty$. This result has potential to be converted to an asymptotic evaluation of the $2n$'th moment of the standard $L$-values for $\mathrm{GL}(n)$. I will describe the main points of the proof which uses spectral decomposition, integral representation of $L$-functions, regularization of Eisenstein series, and use of analytic newvectors for $\mathrm{GL}_n(\mathbb{R})$.
The link for the talk is https://zoom.us/j/94752830725, the password is the order of $\mathrm{SL}_2(\mathbb{F}_{97})$.