L-functions and equations

该提案的第一个主要主题是研究亏格1的曲线上的点，更具体地说，是定义在地面域的可解扩张上的点。这是与Ciperiani的合作，目的是证明对于定义在全实数域上的亏格1曲线，总有点定义在基域的可解扩张上。在第一个例子中，我们的目标是证明这样的点的存在性，同时假设所有p-ady域上都存在点，但这个条件应该很容易消除。我们注意到，我们不希望对曲线的雅可比阶作任何假设。提案的第二个主题是继续研究无法解决的基数变化问题。目前的目的是将注意力限制在GL(2)上的全纯形式的情况。在这个项目中有两个主要问题正在调查中。第一个问题是试图推广十九世纪用来理解方程的思想。对于一元方程，Abel和Galois证明了五次或五次以上的方程不像二次方程那样具有简单的一般公式。这是因为四次或四次以下方程的解总是存在于所谓的可解扩张中，即通过连续提取平方根、立方根等获得的那些扩张。令人惊讶的是，不知道含有一个以上变量的方程是否也是如此。这个项目的第一个目标是证明许多更简单的二元方程确实有这些简单的解。目前数论中的许多构造都需要假设所涉及的方程确实有这样简单的解。该项目的第二部分是试图将其中一种特殊而非常重要的结构推广到所涉及的方程没有任何简单解的情况。

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.