The average rank of elliptic curves. I. (With an appendix by Oisín McGuinness: The explicit formula for elliptic curves over function fields)
type: Item
instance of: scholarly article
title: The average rank of elliptic curves. I. (With an appendix by Oisín McGuinness: The explicit formula for elliptic curves over function fields)
zbMATH Open document ID: 0783.14019
DOI: 10.1007/BF01232033
author: Armand Brumer
published in: Inventiones Mathematicae
publication date: 1993-06-29T00:00:00Z
full work available at URL: https://eudml.org/doc/144027
review text: 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.
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 220743
zbMATH Keywords: Hasse-Weil \(L\)-functions
zbMATH Keywords: average analytic rank of elliptic curves
zbMATH Keywords: height
zbMATH Keywords: Taniyama-Weil conjecture
zbMATH Keywords: ranks of the Mordell-Weil groups
zbMATH Keywords: Birch--Swinnerton-Dyer conjecture
Wikidata QID: Q61440673
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 220743