The Weil representation and Hecke operators for vector valued modular forms

We define Hecke operators for vector valued modular forms transforming with the Weil representation associated to a discriminant form. We describe the properties of the corresponding algebra of Hecke operators and study the action on modular forms.     

Siegel modular forms and representations

This paper explicitly describes the procedure of associating an automorphic representation of PGSp(2n,?) with a Siegel modular form of degree n for the full modular group Γ _n =Sp(2n,ℤ), generalizing the well-known procedure for n=1. This will show that the so-called “standard” and ldquo;spinor”L-functions associated with such forms are obtained as Langlands L-functions. The theory of Euler products, developed by Langlands, applied to a Levi subgroup of the exceptional group of type F _<4, is then used to establish meromorphic continuation for the spinor L-function when n=3. Received: 28 March 2000 / Revised version: 25 October 2000     

Congruences between Siegel modular forms

Using the moduli theory of abelian varieties and a recent result of Böcherer-Nagaoka on lifting of the generalized Hasse invariant, we show congruences between the weights of Siegel modular forms with congruent Fourier expansions. This result implies that the weights of p-adic Siegel modular forms are well defined.     

Galois representations for holomorphic Siegel modular forms

We prove local–global compatibility (up to a quadratic twist) of Galois representations associated to holomorphic Hilbert–Siegel modular forms in many cases (induced from Borel or Klingen parabolic), and as a corollary we obtain a conjecture of Skinner and Urban. For Siegel modular forms, when the local representation is an irreducible principal series we get local–global compatibility without a twist. We achieve this by proving a version of rigidity (strong multiplicity one) for GSp(4) using, on the one hand the doubling method to compute the standard L-function, and on the other hand the explicit classification of the irreducible local representations of GSp(4) over p-adic fields; then we use the existence of a globally generic Hilbert–Siegel modular form weakly equivalent to the original and we refer to Sorensen (Mathematica 15:623–670, 2010) for local–global compatibility in that case.     

Linear dependence among Siegel modular forms

Theorems are given which describe when high enough vanishing at the cusps implies that a Siegel modular cusp form is zero. Formerly impractical computations become practical and examples are given in degree four. Vanishing order is described by kernels, a type of polyhedral convex hull. Received: November 19, 1998 / Revised: July 5, 1999 / Published online: September 5, 2000     

A note on Siegel modular forms

This paper gives a new identification for Siegel modular forms with respect to any congruence subgroup by investigating the properties of their Fourier-Jacobi expansions, and verifies a comparison theorem for the dimensions of the spaces Skn (Γn) and J0k, 1 (Γn) with small weight k. These results can be used to estimate the dimension of the space of modular forms.     

Congruences between Siegel modular forms II

Using a p-adic monodromy theorem on the affine ordinary locus in the minimally compactified moduli scheme modulo powers of a prime p of abelian varieties, we extend Katz?s results on congruence and p-adic properties of elliptic modular forms to Siegel modular forms of higher degree.     

Congruences for Siegel modular forms and their weights

J.-P. Serre proved that the congruences for elliptic modular forms mod p ^m descend to those of weights mod p ^m−1(p−1). Later, this result was generalized by T. Ichikawa to the case of Siegel modular forms. In this note we use elementary methods to reduce Ichikawa’s result to a similar question about elliptic modular forms with level, where results of Katz are available.     

Congruences for Hecke eigenvalues of Siegel modular forms

We prove some congruences for Hecke eigenvalues of Klingen-Eisenstein series and those of cusp forms for Siegel modular groups modulo special values of automorphic L-functions.     

The rationality of vector valued modular forms associated with the Weil representation

 In a recent paper [Duke Math. J., 97, 219–233], Borcherds asks whether or not the spaces of vector valued modular forms associated to the Weil representation have bases of modular forms whose Fourier expansions have only integer coefficients. We give an affirmative answer to Borcherds' question. This strengthens and simplifies Borcherds' main theorem which is a generalization of a theorem of Gross, Kohnen, and Zagier [Math. Ann., 278, 497–562]. Received: 27 September 2001 / Revised version: 22 July 2002 / Published online: 28 March 2003 Mathematics Subject Classification (1991): 11F30; 11F27     

A cohomology for vector valued differential forms

A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Frölicher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is functorial under local diffeomorphisms. This cohomology is determined as the direct product of the de Rham cohomology space and the graded Lie algebra of traceless vector valued differential forms, equipped with a new natural differential concomitant as graded Lie bracket. We find two graded Lie algebra structures on the space of differential forms. Some consequences and related results are also discussed.     

The Clark-Ocone formula for vector valued Wiener functionals

The classical representation of random variables as the Itô integral of nonanticipative integrands is extended to include Banach space valued random variables on an abstract Wiener space equipped with a filtration induced by a resolution of the identity on the Cameron-Martin space. The Itô integral is replaced in this case by an extension of the divergence to random operators, and the operators involved in the representation are adapted with respect to this filtration in a suitably defined sense.A complete characterization of measure preserving transformations in Wiener space is presented as an application of this generalized Clark-Ocone formula.