Published on *Max Planck Institute for Mathematics* (https://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Alexei Pantchichkine
Affiliation:

Université Grenoble Alpes / MPIM
Date:

Wed, 2019-02-27 14:30 - 15:30 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 via {\it admissible measures of Amice-V\'elu}, see also \cite{MTT}. The $p$-ordinary case was treated in \cite{EHLS} via algebraic geometry (method of Katz). 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 $L$ values in the form of certain integrals with respect to Mazur-type measures, saisfying Coates-Perrin-Riou conjectures. Let us call such Euler product weakly motivic at $p$. Both Rankin-Selberg and doubling methods are used.

**Links:**

[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] https://www.mpim-bonn.mpg.de/node/3444

[3] https://www.mpim-bonn.mpg.de/node/246