北京雁栖湖应用数学研究院

A web of related conjectures, due to works of Goldfeld, Katz--Sarnak, Poonen-Rains, and Bhargava--Kane--Lenstra--Poonen--Rains, predict the distributions of ranks and Selmer groups of elliptic curves over Q. However the data agrees quite poorly with these predictions: on average, the ranks appear to be bigger in the data, while the 2-Selmer groups appear to be smaller. In this talk, we will discuss joint work with Takashi Taniguchi, in which we give a theoretical explanation for deviation of the data on 2-Selmer groups from the predicted distribution, namely, the existence of a secondary term.

The cokernel of a random $p$-adic matrix can be used to study the distribution of objects that arise naturally in number theory. For example, Cohen and Lenstra suggested a conjectural distribution of the p-parts of the ideal class groups of imaginary quadratic fields. Friedman and Washington proved that the distribution of the cokernel of a random $p$-adic matrix is the same as the Cohen–Lenstra distribution. Wood generalized the result of Friedman–Washington by considering a far more general class of measure on $p$-adic matrices. In this talk, we explain a further generalization of Wood’s work. This is joint work with Dong Yeap Kang and Jungin Lee.

In this talk, we first give a brief introduction to the Cohen-Lenstra Heuristics which predict the distribution of the p-part of class groups. However the method only works for primes that do not divide the order of the Galois group, or just the so-called ``good primes'' in general. Then we show by examples and some results that when it comes to bad primes, the statistical behavior of the p-part of class groups is qualitatively different from a random module in the heuristics. And we introduce its connection with Gerth's conjecture, which could be described as generalizing Cohen-Lenstra Heuristics to bad primes.<br>

We prove the distribution of Selmer ranks in the certain family of $p$-th twists of CM abelian varieties obeys the symplectic distribution or the unitary distribution. As an application, for a prime $p\geq 3$, we obtain that the twisted Fermat curve $X^p+Y^p=\delta$ over a number field containing a primitive $p$-th root of unity is ``largely" unsolvable as $\delta$ varies.