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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Upper bounds for the number of number fields with alternating Galois group


Authors: Eric Larson and Larry Rolen
Journal: Proc. Amer. Math. Soc. 141 (2013), 499-503
MSC (2010): Primary 11R99, 11G99
Published electronically: June 25, 2012
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Abstract: Let $ N(n, A_n, X)$ be the number of number fields of degree $ n$ whose Galois closure has Galois group $ A_n$ and whose discriminant is bounded by $ X$. By a conjecture of Malle, we expect that $ N(n, A_n, X)\sim C_n\cdot X^{\frac {1}{2}} \cdot (\log X)^{b_n}$ for constants $ b_n$ and $ C_n$. For $ 6 \leq n \leq 84393$, the best known upper bound is $ N(n, A_n, X) \ll X^{\frac {n + 2}{4}}$, by Schmidt's theorem, which implies there are $ \ll X^{\frac {n + 2}{4}}$ number fields of degree $ n$. (For $ n > 84393$, there are better bounds due to Ellenberg and Venkatesh.) We show, using the important work of Pila on counting integral points on curves, that $ N(n, A_n, X) \ll X^{\frac {n^2 - 2}{4(n - 1)}+\epsilon }$, thereby improving the best previous exponent by approximately $ \frac {1}{4}$ for $ 6 \leq n \leq 84393$.


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Additional Information

Eric Larson
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: elarson3@gmail.com

Larry Rolen
Affiliation: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email: larry.rolen@mathcs.emory.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11543-6
PII: S 0002-9939(2012)11543-6
Received by editor(s): July 6, 2011
Published electronically: June 25, 2012
Additional Notes: The authors are grateful for the support of the NSF in funding the Emory 2011 REU. The authors would like to thank our advisor, Andrew Yang, as well as Ken Ono for their guidance, useful conversations, improving the quality of exposition of this article, and hosting the REU
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.