Arithmetic Statistics: Groups of Elliptic Curves and Abelian Varieties, and Zeroes of Families of Curves over Finite Fields.

本研究计划涉及算术统计学，这是数论的一个新的分支，它产生于过去几十年有关数论对象的分布问题的引人注目的新发展。我们正在研究的两个主要分布问题是：（1）如果我们看一个随机的群序列，如二次虚扩张的类群，或有限域上椭圆曲线的点群，这些群是如何分布的？一个简单的概率模型是相同大小的组是均匀分布的，但强有力的证据表明，概率与自同构组的大小成反比，正如科恩-伦斯特拉统计学所预测的那样。(2)有限域上曲线族中点的数目分布的统计量是什么？当研究有限域的大小变大的家庭中的这种变化时，分布来自Katz和Sarnak证明的随机矩阵理论。相比之下，当有限域的大小达到大亏格极限时，对于族的研究，人们应该期望什么，并没有一个普遍的理论图景。

This research proposal is concerned with Arithmetic Statistics, a new branch of numbertheory which emerged from the spectacular new developments of the last decadesabout distribution questions related to number theoritic objects. The two maindistribution questions that we are investigating are:(1)If we look at a random sequence of groups,such as class groups of quadratic imaginary extensions, or groups ofpoints of elliptic curves over finite fields, how are those groups distributed?A naive probabilistic model would be that groups of the same size are uniformlydistributed, but strong evidence indicates that the probability is inversely proportional to the size of the automorphism group, as predicted by the Cohen-Lenstra heuristics.(2) What are the statistics for the distribution of the number of points in families of curves over finite fields? When studying this variation in families where the size of the finite field gets large, the distribution is coming from random matrix theory as proven by Katz and Sarnak. By contrast, there is no general conjectural picture of what one should expect for the study of families at the large genus limit when the size of the finite field.