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SUMMARY:P-adic L-functions for GL(3) - Chris Williams (University of Warwi
 ck)
DTSTART:20211109T143000Z
DTEND:20211109T153000Z
UID:TALK162202@talks.cam.ac.uk
CONTACT:Rong Zhou
DESCRIPTION: Let $\\pi$ be a $p$-ordinary cohomological cuspidal automorph
 ic representation of $GL_n(\\mathbb{A}_\\mathbb{Q})$. A conjecture of Coat
 es--Perrin-Riou predicts that the (twisted) critical values of its $L$-fun
 ction $L(\\pi \\times \\chi\,s)$\, for Dirichlet characters $\\chi$ of $p$
 -power conductor\, satisfy systematic congruence properties modulo powers 
 of $p$\, captured in the existence of a $p$-adic $L$-function. For $n = 1\
 ,2$ this conjecture has been known for decades\, but for $n \\geq 3$ it is
  known only in special cases\, e.g. symmetric squares of modular forms\; a
 nd in all known cases\, $\\pi$ is a functorial transfer from a proper subg
 roup of $GL_n$. I will explain what a $p$-adic $L$-function is\, state the
  conjecture more precisely\, and then report on ongoing joint work with Da
 vid Loeffler\, in which we prove this conjecture for $n=3$ (without any tr
 ansfer or self-duality assumptions)
LOCATION:MR13
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