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，这是数论的一个新分支，是近几十年来关于与数论对象有关的分布问题的惊人新发展中出现的。我们正在研究的两个主要的分布问题是：****(1)如果我们观察一个随机的群序列，比如二次虚扩展的类群，或者有限域上椭圆曲线的点群，这些群是如何分布的？一个朴素的概率模型会认为相同大小的群体是均匀分布的，但强有力的证据表明，概率与自同构群体的大小成反比，正如Cohen-Lenstra启发式所预测的那样。****(2)有限域上曲线族中点数分布的统计数据是什么？当在有限域的大小变大的家族中研究这种变化时，其分布来自于Katz和Sarnak证明的随机矩阵理论。相比之下，对于在大属极限处的科的研究，当有限域的大小增大时，人们应该期待什么，并没有一个一般的推测图

This research proposal is concerned with Arithmetic Statistics, a new branch of number**theory which emerged from the spectacular new developments of the last decades**about distribution questions related to number theoritic objects. The two main**distribution 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 of**points of elliptic curves over finite fields, how are those groups distributed?**A naive probabilistic model would be that groups of the same size are uniformly**distributed, 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.**