The average rank of elliptic curves. I. (With an appendix by Oisín McGuinness: The explicit formula for elliptic curves over function fields)
There has been much interest in the question of what to ``expect'' the rank of an elliptic curve over \(\mathbb{Q}\) to be. The traditional philosophy is that most elliptic curves would have as small of a rank as allowed by the sign of their functional equations, namely either 0 or 1. More recent work has uncovered families of elliptic curves with somewhat higher rank than this philosophy predicts. -- This paper studies the average analytic rank of elliptic curves over \(\mathbb{Q}\) (as given by their Hasse-Weil \(L\)- functions), modulo several standard conjectures. The elliptic curves are ordered, essentially by their Faltings' height. The paper shows that the average rank is then at most 2.3. The techniques of this paper are analytic making use of the explicit formulae. It assumes the Taniyama-Weil conjecture and a form of generalized Riemann hypothesis. For the results to apply to the ranks of the Mordell-Weil groups of the elliptic curves, one needs to assume the weak Birch and Swinnerton-Dyer conjecture as well. While these conjectures are substantial, they are generally expected to be true.
average analytic rank of elliptic curves
scientific article; zbMATH DE number 220743
scientific article