L-functions and equations

点击翻译按钮获取中文摘要

The first main theme of the proposal is the study of points on curves of genus 1, more particularly points which are defined over solvable extensions of the ground field. This is joint work with Ciperiani and aims to prove that for genus 1 curves defined over a totally real number field there are always points defined over a solvable extension of the base field. In the first instance we are aiming to prove the existence of such points while assuming that there exist points over all p-adic fields, but this condition should be easy to remove. We note that we do not wish to assume anything about the rank of the Jacobian of the curve. The second theme of the proposal is to continue a study of the problem of non-solvable base change. For the moment the intent is to restrict attention to the case of holomorphic forms on GL(2). There are two main problems being investigated in this project. The first problem is to try to extend the ideas used in the nineteenth century to understand equations. For equations of one variable it was shown by Abel and Galois that those of degree five or more do not have easy general formulas like the ones for quadratic equations. The reason for this was that the solutions to equations of degree four or less always live in what are called solvable extensions i.e. those extensions obtained by successively extracting square roots, cube roots etc. Surprisingly it is not known whether the same is true for equations with more than one variable. The first aim of this project is to show that many of the simpler kinds of equations in two variables do in fact have these simple kinds of solutions. Many constructions in number theory at the moment require the assumption that the equations involved do have such simple solutions. The second part of the project is an attempt to extend a particular and very important one of these constructions to the situationwhere the equations involved do not have any simple solutions.