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SUMMARY:Shih-Yu Chen (Academia Sinica)
DTSTART:20220316T023000Z
DTEND:20220316T033000Z
DTSTAMP:20260423T022839Z
UID:POINTS/28
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/POINTS/28/">
 On Deligne's conjecture for critical values of tensor product L-functions 
 and symmetric power L-functions of modular forms</a>\nby Shih-Yu Chen (Aca
 demia Sinica) as part of POINTS - Peking Online International Number Theor
 y Seminar\n\n\nAbstract\nIn this talk\, we introduce our result on the alg
 ebraicity of ratios of product of critical values of Rankin--Selberg $L$-f
 unctions and its applications. \nMore precisely\, let $\\mathit{\\Sigma\,\
 \Sigma'}$ (resp. $\\mathit{\\Pi\,\\Pi'}$) be cohomological tamely isobaric
  automorphic representations of $\\mathrm{GL}_n(\\mathbb{A})$ (resp. $\\ma
 thrm{GL}_{n'}(\\mathbb{A})$) such that $\\mathit{\\Sigma}_\\infty = \\math
 it{\\Sigma}_\\infty'$ and $\\mathit{\\Pi}_\\infty = \\mathit{\\Pi}_\\infty
 '$. It is a consequence of Deligne's conjecture on critical $L$-values tha
 t the ratio \n\\[\n\\frac{L(s\, \\mathit{\\Sigma} \\times \\mathit{\\Pi}) 
 \\cdot L(s\,\\mathit{\\Sigma}' \\times \\mathit{\\Pi}')}{L(s\,\\mathit{\\S
 igma} \\times \\mathit{\\Pi}')\\cdot L(s\,\\mathit{\\Sigma}' \\times \\mat
 hit{\\Pi})}\n\\]\nis algebraic and Galois-equivariant at critical points.\
 nWe show that this assertion holds under certain parity and regularity con
 ditions.\nAs applications\, we prove Deligne's conjecture for some tensor 
 product $L$-functions and symmetric odd power $L$-functions for $\\mathrm{
 GL}_2$.\n\nZoom ID：817 4314 0004\n\nZoom password：199319\n
LOCATION:https://researchseminars.org/talk/POINTS/28/
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