David S. Yuen

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

Projectile Motion with Resistance and the Lambert W Function

A

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

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.

TOWARDS THE SIEGEL RING IN GENUS FOUR

Ed Packel (packel@lakeforest.edu) did his undergraduate work at Amherst College and received a Ph.D. in functional analysis from M.I.T. in 1967. Since 1971 he has taught at Lake Forest College, where he served as department chair from 1986 to 1996. His research interests have oscillated among functional analysis, game theory, social choice theory, information-based complexity, and the use of technology (Mathematica) in teaching. His recreational enthusiasms have somehow gravitated towards sports where low numbers are good-namely, competitive distance running and golf.

The extreme core

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.

Restriction of Siegel modular forms to modular curves

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

Estimates for Dimensions of Spaces of Siegel Modular Cusp Forms

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.

LIFTING PUZZLES IN DEGREE FOUR

Runge gave the ring of genus three Siegel modular forms as a quotient ring, R3/〈J(3)〉. R3 is the genus three ring of code polynomials and J(3) is the difference of the weight enumerators for the e8 ⊕ e8 and d+16 codes. Freitag and Oura gave a degree 24 relation, R (4) 0 , of the corresponding ideal in genus four; R (4) 0 is also a linear combination of weight enumerators. We take another step toward the ring of Siegel modular forms in genus four. We explain new techniques for computing with Siegel modular forms and actually compute six new relations, classifying all relations through degree 32. We show that the local codimension of any irreducible component defined by these known relations is at least 3 and that the true ideal of relations in genus four is not a complete intersection. Also, we explain how to generate an infinite set of relations by symmetrizing first order theta identities and give one example in degree 32. We give the generating function of R5 and use it to reprove results of Nebe [25] and Salvati Manni [37]. §