Number Theory Seminar
Tristan Phillips
Dartmouth College
Counting Elliptic Curves Over Number Fields
Abstract: Let $E$ be an elliptic curve over a number field $K$. By the Mordell-Weil Theorem, the set of rational points $E(K)$ forms a finitely generated abelian group, which can be expressed as $E(K) \cong E(K)_\text{tors} \times \mathbb{Z}^r$, where $E(K)_{\text{tors}}$ is the finite torsion subgroup and $r$ is the rank of $E$. In this talk, I will present results on the frequency with which elliptic curves exhibit a prescribed torsion subgroup, and how to establish bounds on the average analytic rank of elliptic curves over number fields. A key approach underlying these results involves employing techniques from Diophantine geometry to count points of bounded height on genus zero modular curves.
Friday October 18, 2024 at 9:00 AM in 636 SEO