Jerry Wang, Department of Pure Mathematics, University of Waterloo
Conjectures on the statistics of elliptic curves are generally formulated with the assumption that the curves in question are ordered by their conductors. However, when proving results on the statistics of elliptic curves, the curves are usually ordered by (naive) height. There are two main reasons for doing so: first, it is difficult to rule out the possibility that there are many elliptic curves with large height but small discriminant; second, it is difficult to rule out the possibility that there are many elliptic curves with large discriminant but small conductor. In this talk, we will focus on the second question. We assume the j-invariant is bounded and count the number of elliptic curves having bounded conductor and whose discriminant is bounded by their conductor to the power 7/4. We also count elliptic curves where the quotient of discriminant by conductor is squarefree and prove that the average size of their 2-Selmer groups is 3.
This is joint work with Ananth Shankar and Arul Shankar.
MC 5417