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

一个方程有几个解？答案不仅取决于所讨论的方程式，还取决于发问的人。研究彗星运动方程的物理学家可能会问，彗星在太空中可以走多少种不同的路径(轨道是周期性的吗？)另一方面，数学家对丢番图方程感兴趣。这些方程具有一些变量和整数(即整数)系数，研究时要理解，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：一个丢番图方程有多少个解？解丢番图方程是困难的，研究一族方程并看看关于这一族的大规模行为(解的数量是有限的还是无限的？)通常是很有趣的。在过去的10年里，Manjul Bhargava和他的合作者采用了这种观点，并证明了数论中的一些开创性的结果。他们特别关注由椭圆和超椭圆曲线族产生的丢番图方程族。它们表明，被称为Selmer群的解集合的某些代理可以与射影空间中的二进制四次形式和二次曲线的铅笔等具体的经典对象相关。在他们用这种方法证明的许多定理中，让我们只提到椭圆曲线在有理数上只有有限多(有理)解的正比例存在--一个引人注目的定性结果。我们将研究非超椭圆曲线族的算法。我们感兴趣的族来自形变理论和代数几何，它们是例外曲线奇点的逆变形。超椭圆曲线是最简单的代数曲线，是射影直线的二重覆盖，其几何和算法相对容易理解。我们的非超椭圆族的算术就不那么明确了。我们将利用我们的家族与形变理论和表示理论的隐藏联系来获得关于Selmer群的精确和完整的结果，就像Bhargava和他的合作者在超椭圆情况下的结果一样。

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.