Paramodular cusp forms

Abstract

We classify Siegel modular cusp forms of weight two for the paramodular group K(p) for primes p < 600. We find that weight two Hecke eigenforms beyond the Gritsenko lifts correspond to certain abelian varieties defined over Q of conductor p. The arithmetic classification is in the companion article by A. Brumer and K. Kramer [4]. The Paramodular Conjecture, supported by these computations and consistent with the Langlands philosophy and the work of H. Yoshida [31], is a partial extension to degree 2 of the Shimura-Taniyama Conjecture. These nonlift Hecke eigenforms share Euler factors with the corresponding abelian variety A and satisfy congruences modulo ℓ with Gritsenko lifts, whenever A has rational ℓtorsion.

Borcherds Products Everywhere

“…In [21] it was shown that for primes p < 600, if p ∈ {277, 349, 353, 389, 461, 523, 587} then the weight two paramodular cusp forms are spanned by Gritsenko lifts, that is, …”

Theta lifts of Bianchi modular forms and applications to paramodularity

“…In [5], there are examples of abelian surfaces of prime conductor. The first few Euler factors for each of them are shown to match those of a paramodular form in [27]. However, none of those surfaces has been proved to be modular.…”

Antisymmetric paramodular forms of weight 3

“…Antisymmetric paramodular cusp forms of weight 2 are also very interesting. For a prime polarisation, such a form might exist only for p = 587 (see [26]). At the moment, only three examples are known (see [19]) for t = 587, 713, 893.…”