On Deligne's conjecture for critical values of tensor product L-functions and symmetric power L-functions of modular forms

Abstract: In this talk, we introduce our result on the algebraicity of ratios of product of critical values of Rankin--Selberg $L$-functions and its applications. More precisely, let $\mathit{\Sigma,\Sigma'}$ (resp. $\mathit{\Pi,\Pi'}$) be cohomological tamely isobaric automorphic representations of $\mathrm{GL}_n(\mathbb{A})$ (resp. $\mathrm{GL}_{n'}(\mathbb{A})$) such that $\mathit{\Sigma}_\infty = \mathit{\Sigma}_\infty'$ and $\mathit{\Pi}_\infty = \mathit{\Pi}_\infty'$. It is a consequence of Deligne's conjecture on critical $L$-values that the ratio \[ \frac{L(s, \mathit{\Sigma} \times \mathit{\Pi}) \cdot L(s,\mathit{\Sigma}' \times \mathit{\Pi}')}{L(s,\mathit{\Sigma} \times \mathit{\Pi}')\cdot L(s,\mathit{\Sigma}' \times \mathit{\Pi})} \] is algebraic and Galois-equivariant at critical points. We show that this assertion holds under certain parity and regularity conditions. As applications, we prove Deligne's conjecture for some tensor product $L$-functions and symmetric odd power $L$-functions for $\mathrm{GL}_2$.

algebraic geometrynumber theoryrepresentation theory

Audience: researchers in the topic

Comments: Zoom ID：817 4314 0004

Zoom password：199319

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POINTS - Peking Online International Number Theory Seminar

Series comments: Description: Seminar on number theory and related topics

This seminar series is sponsored by the Beijing International Center of Mathematical Research (BICMR) and the School of Mathematical Sciences of Peking University.

The conference number and password are available on the external website. See also the announcements on bicmr.pku.edu.cn

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