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 $\mathrm{GL}_n(\mathbb{R})$.

Zoom ID: 919 6497 4060

For password contact Pieter Moree (moree@mpim...).

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