Cris Poor

Paramodular cusp forms

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 the rationals of conductor p. The arithmetic classification is in a companion article by A. Brumer and K. Kramer. The Paramodular Conjecture, supported by these computations and consistent with the Langlands philosophy and the work of H. Yoshida, is a partial extension to degree 2 of the Shimura-Taniyama Conjecture. These nonlift Hecke eigenforms share Euler factors with the corresponding abelian variety

Dimensions of cusp forms for Γ0(p) in degree two and small weights

A

SCHOTTKY'S FORM AND THE HYPERELLIPTIC LOCUS.

and satisfy congruences modulo \ell with Gritsenko lifts, whenever

Dimensions of spaces of Siegel modular forms of low weight in degree four

A

THE CUSP STRUCTURE OF THE PARAMODULAR GROUPS FOR DEGREE TWO

has rational \ell-torsion.

The hyperelliptic locus

We investigate degree two Siegel cusp forms of small weight for Γ0(p). Using the Restriction Technique we compute some dimensions and verify the conjectures ofHashimoto in some examples of weights three and four. For weight two we determine the dimension for primesp ≤ 41 and find only lifts. We explain in general how to compute spaces of Siegel cusp forms for subgroups of finite index in Γn.

TOWARDS THE SIEGEL RING IN GENUS FOUR

We show that Schottkys modular form, Jg, has in every genus an irreducible divisor which contains the hyperelliptic locus. We also improve a corollary of Igusa concerning Siegel modular forms that must necessarily vanish on the hyperelliptic locus.

The extreme core

We calculate the dimensions of using Erokhins work on Niemeier lattices and geometric methods involving the hyperelliptic locus.

Slopes of integral lattices

We describe the one-dimensional and zero-dimensional cusps of the Satake compactification for the paramodular groups in degree two for arbitrary levels. We determine the crossings of the one-dimensional cusps. Applications to computing the dimensions of Siegel modular forms are given.

Restriction of Siegel modular forms to modular curves

It is shown in theorem 2.6.1 that the vanishing properties of the thetanullwerte of hyperelliptic Jacobians characterize them among all irreducible principally polarized abelian varieties. An alternate proof of Mumford’s theorem characterizing hyperelliptic Jacobians among all principally polarized abelian varieties by vanishing and nonvanishing properties is sketched in section 2.7.