Some Heuristics about Elliptic Curves

Abstract

We give some heuristics for counting elliptic curves with certain properties. In particular, we rederive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to {\small$X$}, which is an application of lattice-point counting. We then introduce heuristics that allow us to predict how often we expect an elliptic curve $E$ with even parity to have $L(E,1)=0$. We find that we expect there to be about $c_1X^{19/24}(\log X)^{3/8}$ curves with $|\Delta|<X$ with even parity and positive (analytic) rank; since Brumer and McGuinness predict {\small$cX^{5/6}$} total curves, this implies that, asymptotically, almost all even-parity curves have rank $0$. We then derive similar estimates for ordering by conductor, and conclude by giving various data regarding our heuristics and related questions.