Open Access
2008 Some Heuristics about Elliptic Curves
Mark Watkins
Experiment. Math. 17(1): 105-125 (2008).


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.


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Mark Watkins. "Some Heuristics about Elliptic Curves." Experiment. Math. 17 (1) 105 - 125, 2008.


Published: 2008
First available in Project Euclid: 18 November 2008

zbMATH: 1151.14025
MathSciNet: MR2410120

Primary: 14G10 , 14H52

Keywords: asymptotic count , Elliptic curves , vanishing $L$-function

Rights: Copyright © 2008 A K Peters, Ltd.

Vol.17 • No. 1 • 2008
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