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The average rank of elliptic curves. I. (With an appendix by Oisín McGuinness: The explicit formula for elliptic curves over function fields). (English) Zbl 0783.14019
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.
Reviewer: J.Jones (Tempe)

14H52 Elliptic curves
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H05 Algebraic functions and function fields in algebraic geometry
11G05 Elliptic curves over global fields
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