Defects of cusp singularities and the classification of Hilbert modular threefolds

This material is based on work supported by the National Science Foundation under Grant No. DMS-9008689. The Government has certain rights in this material     

Computing the arithmetic genus of Hilbert modular fourfolds

The Hilbert modular fourfold determined by the totally real quartic number field is a desingularization of a natural compactification of the quotient space , where PSL acts on by fractional linear transformations via the four embeddings of into . The arithmetic genus, equal to one plus the dimension of the space of Hilbert modular cusp forms of weight , is a birational invariant useful in the classification of these varieties. In this work, we describe an algorithm allowing for the automated computation of the arithmetic genus and give sample results.

     

Defect series and nonrational Hilbert modular threefolds

This material is based on work supported by the National Science Foundation under Grant No. DMS-9115349. The Government has certain rights in this material     

Tetranomial Thue equations

Let F(x,y)$F(x,y)$ be an irreducible binary form of degree n?6$n?6$ with exactly four nonzero terms. Assuming certain conditions relating the coefficients and the degrees of the different terms are satisfied, we prove upper bounds on the number of equivalent pairs of nontrivial solutions of the Thue equation |F(x,y)|=1$|F(x,y)|=1$. Improved bounds are provided for a variety of cases, where more information about the form is known.     

Invariants of an augmentation of Γ0(n)

Let G _n ^× be the 2-group of primary factors of a positive integer n and fix a direct product decomposition of this group. We define an augmentation of Γ₀(n) based on G _n ^×, paralleling augmentations used by Fricke, Cohn and Knopp, and others. Using the decomposition of G _n ^×, we then define a family of functions based on η-functions and use these functions to construct invariants of the augmented group. Along with proving results analogous to those of Cohn and Knopp, we make a complete determination of the multiplier systems for these new functions.     

Multipliers of a Family of Almost Modular Functions

The goal of this paper is to complete an investigation begun by Cohn and Knopp in their 1994 paper, Application of Dedekind eta-multipliers to modular equations. The paper concerned _k (z), a family of modular forms on ⁰(N) (N a positive integer) with possibly non-trivial multiplier systems. Cohn and Knopp defined new functions _k (z) and a new group containing ⁰(N) and proved that for all S in the larger group and for all k, _k (Sz) = M _k(S) _k (z), where M _k(S)²⁴ = 1. This yielded interesting invariance properties of _k , dependent on the values of M _k(S). Fixing a constant integer e, independent of k, Cohn and Knopp proved that for all k and all S in the larger group, M _k(S) ^e = (±1) ^e . They determined the sign of M _k(S) ^e in many, but not all, cases. In this paper, we give a complete determination of the values of M _k(S) ^e in the remaining cases.     

Explicit resolutions of cubic cusp singularities

Resolutions of cusp singularities are crucial to many techniques in computational number theory, and therefore finding explicit resolutions of these singularities has been the focus of a great deal of research. This paper presents an implementation of a sequence of algorithms leading to explicit resolutions of cusp singularities arising from totally real cubic number fields. As an example, the implementation is used to compute values of partial zeta functions associated to these cusps.

     

Galois realizability of groups of order 64

This article examines the realizability of groups of order 64 as Galois groups over arbitrary fields. Specifically, we provide necessary and sufficient conditions for the realizability of 134 of the 200 noncyclic groups of order 64 that are not direct products of smaller groups.     

Realizability and automatic realizability of Galois groups of order 32

This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.