Counting Elliptic Curves over Number Fields

Tristan Phillips (University of Arizona)

22-Feb-2022, 21:00-22:00 (2 years ago)

Abstract: In this talk I will discuss some results on counting elliptic curves over number fields. In particular, I will give asymptotics for the number of isomorphism classes of elliptic curves over arbitrary number fields with certain prescribed level structures and prescribed local conditions. This is done by counting the number of points of bounded height on genus zero modular curves which are isomorphic to a weighted projective space. This includes the cases of X(N) for N\in\{1,2,3,4,5\}, X_1(N) for N\in\{1,2,\dots,10,12\}, and X_0(N) for N\in\{1,2,4,6,8,9,12,16,18\}. Using these results for counting elliptic curves over number fields with a prescribed local condition, one can show that the average analytic rank of elliptic curves over any number field K is bounded above by 3\text{deg}(K)+1/2, under the assumptions that all elliptic curves over K are modular and have L-functions which satisfy the Generalized Riemann Hypothesis

number theory

Audience: researchers in the topic


University of Arizona Algebra and Number Theory Seminar

Organizers: Aparna Upadhyay*, Pan Yan*
*contact for this listing

Export talk to