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Bulletin of the American Mathematical Society

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Average ranks of elliptic curves: Tension between data and conjecture

Authors: 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
Published electronically: February 15, 2007
MathSciNet review: 2291676
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Abstract: 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

William Stein
Affiliation: (W. Stein) Department of Mathematics, University of Washington, Seattle, Washington 98195

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
Article copyright: © Copyright 2007 Baur Bektemirov, Barry Mazur, William Stein, and Mark Watkins

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