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Admissible measures for Hermitian modular forms

Posted in
Alexei Pantchichkine
University Grenoble Alpes/MPI
Tue, 2017-07-25 14:00 - 15:00
MPIM Lecture Hall

For a prime  $p$ and a positive integer  $n$, the standard zeta function $L_F(s)$ is considered,
attached to an Hermitian modular form  $F=\sum_H A(H) q^H$ on the Hermitian upper half plane
$\mathcal H_m$  of degree $n$, where $H$ runs through semi-integral positive definite Hermitian
matrices of degree $n$, i.e. $H\in \Lambda_m({\mathcal O})$ over the integers ${\mathcal O}$
of an imaginary quadratic field $K$, where $q^H=\exp(2\pi i {\rm Tr}(HZ))$.
Analytic $p$-adic continuation of their zeta functions is constructed via   $p$-adic measures, bounded
or growing. Previously this problem was solved for the Siegel modular forms.
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 $L_F(s)$ 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.

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