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

这项研究涉及算术统计学，这是数论的一个新分支**，它是在过去几十年**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**分布问题是：*(1)如果我们观察随机的群序列，**例如二次虚扩张的类群，或者有限域上的椭圆曲线的**点群，这些群是如何分布的？**一个朴素的概率模型是相同大小的群是均匀分布的，但强有力的证据表明，概率与自同构群的大小成反比，正如Cohen-Lenstra启发式所预测的那样。*(2)有限域上曲线族中点的数目的分布的统计数字是什么？当研究有限域规模变大的家庭中的这种变化时，分布来自随机矩阵理论，正如Katz和Sarnak所证明的那样。相比之下，当有限域的大小时，对于在大亏格极限上的科的研究应该期待什么，没有一般的猜想图景。

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.**