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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Average ranks of elliptic curves: Tension between data and conjecture
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by Baur Bektemirov, Barry Mazur, William Stein and Mark Watkins PDF
Bull. Amer. Math. Soc. 44 (2007), 233-254


Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational point in just the right context, at one place in the theory, can inhibit and control—thanks to ideas of Kolyvagin—the existence of rational points and other mathematical structures elsewhere. Despite all that we know about these objects, the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today. We have a network of heuristics and conjectures regarding rational points, and we have massive data accumulated to exhibit instances of the phenomena. Generally, we would expect that our data support our conjectures, and if not, we lose faith in our conjectures. But here there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either. We discuss various aspects of this story, including recent heuristics and data that attempt to resolve this mystery. We shall try to convince the reader that, despite seeming discrepancy, data and conjecture are, in fact, in harmony.
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Additional Information
  • Baur Bektemirov
  • Affiliation: (B. Bektemirov) Harvard College, Cambridge, Massachusetts 02138
  • Barry Mazur
  • Affiliation: (B. Mazur) Department of Mathematics, Harvard University, Cambridge, Massachu- setts 02138
  • MR Author ID: 121915
  • ORCID: 0000-0002-1748-2953
  • William Stein
  • Affiliation: (W. Stein) Department of Mathematics, University of Washington, Seattle, Washington 98195
  • MR Author ID: 679996
  • Mark Watkins
  • Affiliation: (M. Watkins) School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW England
  • Received by editor(s): January 1, 2006
  • Published electronically: February 15, 2007
  • © Copyright 2007 Baur Bektemirov, Barry Mazur, William Stein, and Mark Watkins
  • Journal: Bull. Amer. Math. Soc. 44 (2007), 233-254
  • MSC (2000): Primary 11-02, 11D25, 11Y35, 11Y40
  • DOI:
  • MathSciNet review: 2291676