Experimental Mathematics

Some Heuristics about Elliptic Curves

Mark Watkins

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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.

Article information

Experiment. Math., Volume 17, Issue 1 (2008), 105-125.

First available in Project Euclid: 18 November 2008

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx] 14G10: Zeta-functions and related questions [See also 11G40] (Birch- Swinnerton-Dyer conjecture)

Elliptic curves asymptotic count vanishing $L$-function


Watkins, Mark. Some Heuristics about Elliptic Curves. Experiment. Math. 17 (2008), no. 1, 105--125. https://projecteuclid.org/euclid.em/1227031901

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