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Arithmetic of non-hyperelliptic curves: rational points via representation theory

非超椭圆曲线的算术：通过表示论的有理点

基本信息

批准号：
EP/N007204/1
负责人：
Jack Thorne
金额：
$ 12.52万
依托单位：
University of Cambridge
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN007204%2F1
关键词：
Arithmetic non hyperelliptic curves rational

项目摘要

How many solutions does an equation have? The answer depends not only on the equation in question but also the person asking. A physicist studying the equations of motion of comets might ask how many different kinds of path a comet can take through space (is the orbit periodic?). On the other hand, number theorists are interested in diophantine equations. These are equations in some number of variables and with integer (i.e. whole number) coefficients, studied with the understanding that one is interested only in solutions where the variables themselves take integer values. We will study the question: how many solutions does a diophantine equation have?Solving diophantine equations is difficult, and it is often interesting to study families of equations and see what can be said about the large-scale behaviour of the family (is the number of solutions finite or infinite?). In the last 10 years Manjul Bhargava and his collaborators have taken this perspective and proved a number of groundbreaking results in number theory. They have focused in particular on families of diophantine equations arising from families of elliptic and hyperelliptic curves. They show that certain proxies for the set of solutions, called Selmer groups, can be related to such concrete and classical objects as binary quartic forms and pencils of quadrics in projective space. Among the many theorems they have proved this way, let us just mention the existence of a positive proportion of elliptic curves over the rationals with only finitely many (rational) solutions -- a striking qualitative result.We will study the arithmetic of families of non-hyperelliptic curves. The families we are interested in come from deformation theory and algebraic geometry, being the versal deformations of exceptional curve singularities. Hyperelliptic curves are the simplest algebraic curves, being double covers of the projective line, and their geometry and arithmetic is relatively accessible. The arithmetic of our non-hyperelliptic families is much less explicit. We will exploit the hidden connections that our families have to deformation theory and representation theory to obtain results about Selmer groups as precise and complete as those of Bhargava and his collaborators in the hyperelliptic case.

一个方程有多少个解？答案不仅取决于问题中的方程式，也取决于提问的人。研究彗星运动方程的物理学家可能会问，彗星在太空中有多少种不同的路径（轨道是周期性的吗？）另一方面，数论学家对丢番图方程感兴趣。这些方程有一定数量的变量和整数（即整数）系数，研究的理解是，人们只对变量本身取整数值的解感兴趣。我们将研究这个问题：丢番图方程有多少个解？求解丢番图方程是困难的，而研究方程组族并看看该族的大尺度行为（解的数量是有限的还是无限的？）在过去的10年里，Manjul Bhargava和他的合作者采用了这一观点，并证明了数论中一些开创性的结果。他们特别关注由椭圆曲线族和超椭圆曲线族产生的丢番图方程族。他们表明，解集的某些代理，称为Selmer群，可以与射影空间中的二元四次形式和二次型铅笔等具体和经典对象相关。在他们以这种方式证明的许多定理中，让我们只提到椭圆曲线在只有有限多个（有理）解的有理数上的正比例的存在性——一个惊人的定性结果。我们将研究非超椭圆曲线族的算法。我们感兴趣的族来自变形理论和代数几何，是异常曲线奇点的通用变形。超椭圆曲线是最简单的代数曲线，是射影线的双重覆盖，它的几何和算术是比较容易理解的。非超椭圆族的算法就不那么明确了。我们将利用我们的家族与变形理论和表征理论之间的隐藏联系，以获得关于Selmer群的结果，就像Bhargava和他的合作者在超椭圆情况下的结果一样精确和完整。