Data di Pubblicazione:
2005
Citazione:
Hilbert modular forms : mod $p$ and $p$-adic aspects / F. Andreatta, E.Z. Goren. - In: MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0065-9266. - 173:819(2005), pp. 1-100.
Abstract:
We study Hilbert modular forms in characteristic~$p$ and over
$p$-adic rings. In the characteristic~$p$ theory we describe the
kernel and image of the $q$-expansion map and prove the existence
of filtration for Hilbert modular forms; we define operators
$U$,~$V$ and~$\Theta_\chi$ and study the variation of the
filtration under these operators. Our methods are geometric --
comparing holomorphic Hilbert modular forms with rational
functions on a moduli scheme with level-$p$ structure, whose poles
are supported on the non-ordinary locus.
In the $p$-adic theory we study congruences between Hilbert
modular forms. This applies to the study of congruences between
special values of zeta functions of totally real fields. It also
allows us to define $p$-adic Hilbert modular forms ``\`a la Serre"
as $p$-adic uniform limit of classical modular forms, and compare
them with $p$-adic modular forms ``\`a la Katz" that are regular
functions on a certain formal moduli scheme. We show that the two
notions agree for cusp forms and for a suitable class of weights
containing all the classical ones. We extend the operators $V$
and~$\Theta_\chi$ to the $p$-adic setting.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Congruence; Filtration; Hilbert modular form; Hilbert modular variety; Zeta function
Elenco autori:
F. Andreatta, E.Z. Goren
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