Number Theory Seminar

How many points does the average elliptic curve have?  Over the rational

numbers this question leads to the study of the distribution of ranks of elliptic

curves. We consider curves ordered by height, a measure of the size of the

coefficients of the defining equation.  We will compare theoretical results and

conjectures to a recently constructed database containing rank and 2-Selmer group

information for all curves of height at most 2.7*10^{10}.  This is joint work with

Balakrishnan, Ho, Spicer, Stein, and Weigandt.

For curves over a fixed finite field we can give precise answers to

statistical questions about the distribution of rational point counts.  For example,

what is the average number of rational points on an elliptic curve over F_q containing a

rational 5-torsion point?  We will discuss joint work with Petrow about these types

of questions, generalizing work of Birch and others.