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.
Let n ≥ 2 be an integer and consider the set Tn of n × n permutation matrices π for which πij = 0 for j ≥ i + 2.In this paper we study the convex hull of Tn, which we denote by Pn. Pn is a polytope of dimension n 2 . Our main purpose is to provide evidence for the following conjecture concerning its volume. Let vn denote the minimum volume of a simplex with vertices in the affine lattice spanned by Tn. Then the volume of Pn is vn times the productWe also give a related result on the Ehrhart polynomial of Pn.
Abstract. We prove the Borcherds Products Everywhere Theorem, Theorem 6.6, that constructs holomorphic Borcherds Products from certain Jacobi forms that are theta blocks without theta denominator. The proof uses generalized valuations from formal series to partially ordered abelian semigroups of closed convex sets. We present nine infinite families of paramodular Borcherds Products that are simultaneously Gritsenko lifts. This is the first appearance of infinite families with this property in the literature.
We complete the program indicated by the Ansatz of D'Hoker and Phong in genus 4 by proving the uniqueness of the restriction to Jacobians of the weight 8 Siegel cusp forms satisfying the Ansatz. We prove dim[ 4 (1, 2), 8] 0 = 2 and dim[ 4 (1, 2), 8] = 7. In each genus, we classify the linear relations among the selfdual lattices of rank 16. We extend the program to genus 5 by constructing the unique linear combination of theta series that satisfies the Ansatz.
Abstract. This article gives upper bounds on the number of FourierJacobi coefficients that determine a paramodular cusp form in degree two. The level N of the paramodular group is completely general throughout. Additionally, spaces of Jacobi cusp forms are spanned by using the theory of theta blocks due to Gritsenko, Skoruppa and Zagier. We combine these two techniques to rigorously compute spaces of paramodular cusp forms and to verify the Paramodular Conjecture of Brumer and Kramer in many cases of low level. The proofs rely on a detailed description of the zero dimensional cusps for the subgroup of integral elements in each paramodular group.
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