Catálogo de publicaciones - libros

The code and some computational results of experiments in number theory are given. The experiments concern covering systems with applications to explicit primality tests, the inverse of Euler’s totient function, and class number relations in Galois extensions of ℚ. Some evidence for various conjectures and open problems is given.

Class field theory of global fields provides a description of finite abelian extensions of number fields and of function fields of transcendence degree 1 over finite fields. After a brief review of the handling of both function and number fields in , we give an introduction to computational class field theory focusing on applications: We show how to construct tables of small degree extensions and how to utilize the class field theory to find curves with many rational points.x

In this article, we will determine the primitive integral solutions to equations of the form

In this paper we describe the Birch and Swinnerton-Dyer conjecture in the case of modular abelian varieties and how to use to do computations with some of the quantities that appear in the conjecture. We assume the reader has some experience with algebraic varieties and number theory, but do not assume the reader has proficiency working with elliptic curves, abelian varieties, modular forms, or modular symbols. The computations give evidence for the Birch and Swinnerton- Dyer conjecture and increase our explicit understanding of modular abelian varieties.

We solve two genus 2 curve problems using . First we give examples of how can be used to find the equation of a genus 2 curve whose Jacobian has prescribed Complex Multiplication. We treat 2 fields, one easy and one harder. Secondly we show how can be used to find, and ultimately prove existence of, rational isogenies between the Jacobians of two genus 2 curves.

Many recent constructions of varieties, including the lists of K3 surfaces in , use graded ring methods.We show how to apply the method using and, as an application, construct 27 families of K3 surfaces that appear as degenerate cases of surfaces in the usual lists. These are displayed in Tables 1–3 and include both standard degenerations and new examples.

In this chapter we construct the split octonion algebra over a ring, first using the structure constant machinery of and then using Lie algebras.

The support variety of a module over a group algebra is an affine variety that encodes many of the homological properties of the module. Although the definition of the support variety is given in terms of the cohomology ring of the group, it can be computed directly from the actions of the elementary abelian subgroups.

A well known theorem of Chouinard states that if is a finite group and is a field of characteristic > 0, then a -module is projective precisely when its restrictions to all elementary abelian -subgroups of are projective. We investigate some similar situations in which the restrictions to subalgebras detect the projectivity of a module over an algebra.

We describe the theory and implementation of some new and more flexible functions for computing cohomology groups of finite groups, and their application to the computation of group extensions.