Some Heuristics about Elliptic Curves

Abstract

Abstract. We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive 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 (with refinements from random matrix theory) 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 (log X) 3/8 curves with |∆| < 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.

Orders of Tate-Shafarevich groups for the Neumann-Setzer type elliptic curves

“…The numerical studies and conjectures by Conrey-Keating-Rubinstein-Snaith [6], Delaunay [11][12], Watkins [33], Radziwi l l-Soundararajan [24] (see also the papers [9] [7] [8] and references therein) substantially extend the systematic tables given by Cremona.…”

“…where c E > 0, and there are four different possibilities for b E , largery dependent on the rational 2-torsion structure of E. Watkins [33], and Park-Poonen-Voight-Wood [22] have conjectured that…”

How many rational points does a random curve have?

“…The Minimalist Conjecture for all elliptic curves and for quadratic twist families is also supported by the philosophy of Katz and Sarnak [KS99] and later random matrix theory computations and heuristics of Keating and Snaith [KS00], Conrey, Keating, Rubinstein, and Snaith [CKRS02], Watkins [Wat08], and others. See [BMSW07,Poo12] for excellent surveys of many aspects of this conjecture.…”

Heuristics for the Arithmetic of Elliptic Curves

“…Let H = ht E. Following Lang [Lan83] (see also [GS95], [dW98], [Hin07], [Wat08], and [HP16]), we estimate the typical size of X 0 by estimating all the other quantities in (2) as H → ∞; see [PPVW16, Section 6] for details. The upshot is that if we average over E and ignore factors that are H o(1) , then (2) simplifies to 1 ∼ X 0 Ω and we obtain X 0 ∼ Ω −1 ∼ H 1/12 .…”