# Weakly motivic Shimura's zeta functions via p-adic L-functions and doubling method

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Speaker:
Alexei Pantchichkine
Affiliation:
Université Grenoble Alpes / MPIM
Date:
Wed, 2019-02-27 14:30 - 15:30
Location:
MPIM Lecture Hall
Parent event:
Number theory lunch seminar
A motivic approach is developped to Shimura's zeta functions and
attached $p$-adic $L$-functions via admissible measures and doubling method.
Shimura's zeta functions $\Zc(s,{\f})$ \cite{Shi00} attached to
holomorphic arithmetical automorphic forms ${\f}$} on unitary groups
$U_K$ of an imaginary quadratic field $K$.
A motivically normalized $L$-function $\Dr(s,{\f})$ attached to
$\Zc(s,{\f})$  is defined in accordance with Deligne  \cite{De79} and
Coates-Perrin-Riou conjectures \cite{CoPe}.
%An explicit description of Shimura's $\Gamma$-factors is used.
The attached $p$-adic $L$-functions  of $\Dr(s,{\f})$ are constructed
The $p$-ordinary case was treated  in \cite{EHLS} via algebraic
The main result is stated in terms of the Hodge polygon $P_{H}(t): [0,d]\to {\mathbb R}$ and the Newton polygon $P_N(t)=P_{N,p}(t): [0,d]\to {\mathbb R}$of the zeta function $\Dr(s,{\f})$ of degree  $d=4n$.
Main theorem gives a $p$-adic analytic interpolation of the $L$
Let us call such Euler product weakly motivic at $p$.
Both Rankin-Selberg and doubling methods are used.