Evento
Descrizione:
Abstract: Let $f$ be a classical modular cuspform of weight $k$ and level $N$ defined over a ring $B$ of $p$-adic integers and let $x$ be a $B$-rational CM-point
in the corresponding modular or Shimura scheme of arithmetic level $N$. When the abelian variety corresponding to $x$ has ordinary reduction mod $p$
we define a power series expansion of $f$ around $x$ with coefficients in some bigger $p$-adic ring $B^\prime$. By Mahler’s theory this power series defines
a measure $\mu_x$ on $\mathbb{Z}_p$ with values in $B^\prime$. By letting $x$ vary in its Galois orbit the measures $\mu_x$ can added together to form a
new measure $\mu$. The squares of the moments of $\mu$ are related via the theory of Harris-Kudla to the special value of a twisted $L$-function
attached to $f$ and the CM field $K$. Moreover, when $f$ is eigen for the Hecke $T_p$-operator we compute the correct interpolation factor for the restriction of
$\mu$ to $\mathbb{Z}_p^\times$.