Hilbert modular forms: mod $p$ and $p$-adic aspects

Details Hilbert modular forms: mod $p$ and $p$-adic aspects Hilbert modular forms: mod $p$ and $p$-adic aspects

2003-08-05

arxiv.org/abs/math/0308040v1

Mathematics

Number Theory

96 pages; Submitted for publication July 30 2001

Scientific paper

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. In particular, we prove that every ordinary eigenform has filtration in a prescribed box of weights. 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 the $p$-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.

Affiliated with

Also associated with

No associations

Rating

Hilbert modular forms: mod $p$ and $p$-adic aspects does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Hilbert modular forms: mod $p$ and $p$-adic aspects, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hilbert modular forms: mod $p$ and $p$-adic aspects will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-571978