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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Almost all elliptic curves are Serre curves


Author: Nathan Jones
Journal: Trans. Amer. Math. Soc. 362 (2010), 1547-1570
MSC (2000): Primary 11G05, 11F80
Published electronically: September 30, 2009
MathSciNet review: 2563740
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Abstract: Using a multidimensional large sieve inequality, we obtain a bound for the mean-square error in the Chebotarev theorem for division fields of elliptic curves that is as strong as what is implied by the Generalized Riemann Hypothesis. As an application we prove that, according to height, almost all elliptic curves are Serre curves, where a Serre curve is an elliptic curve whose torsion subgroup, roughly speaking, has as much Galois symmetry as possible.


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Additional Information

Nathan Jones
Affiliation: Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centreville Station, Montréal, Québec, Canada H3C 3J7
Address at time of publication: Department of Mathematics, University of Mississippi, Hume Hall 305, P.O. Box 1848, University, Mississippi 33677-1848
Email: ncjones@olemiss.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04804-1
PII: S 0002-9947(09)04804-1
Received by editor(s): May 4, 2007
Received by editor(s) in revised form: April 3, 2008
Published electronically: September 30, 2009
Article copyright: © Copyright 2009 American Mathematical Society