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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Upper bounds for the number of number fields with alternating Galois group
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by Eric Larson and Larry Rolen PDF
Proc. Amer. Math. Soc. 141 (2013), 499-503 Request permission

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
  • 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
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 499-503
  • MSC (2010): Primary 11R99, 11G99
  • DOI: https://doi.org/10.1090/S0002-9939-2012-11543-6
  • MathSciNet review: 2996953