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Original Article

Some Heuristics about Elliptic Curves

Pages 105-125 | Published online: 30 Jan 2011
 

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 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 1 X 19/24(logX)3/8 curves with |Δ| < X with even parity and positive (analytic) rank; since Brumer and McGuinness predict 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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