Elliptic curves of low rank

Abstract: For an elliptic curve $E$ over a number field $K$, the set $E(K)$ of $K$-points is a finitely generated abelian group whose rank is an important/mysterious invariant. It is an open and difficult problem to determine which ranks occur for elliptic curves over a fixed number field $K$. We will discuss recent work which shows that there are infinitely many elliptic curves over $K$ of rank $r$ for each integer $0 \leq r \leq 4$. We will construct our curves by specializing well-chosen nonisotrivial families. We will use a result of Kai, which generalizes work of Green, Tao and Ziegler to number fields, to carefully choose our curves in the families.

number theory

Audience: researchers in the topic

BC-MIT number theory seminar