Upper bounds for the number of number fields with alternating Galois group
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- by Eric Larson and Larry Rolen
- Proc. Amer. Math. Soc. 141 (2013), 499-503
- DOI: https://doi.org/10.1090/S0002-9939-2012-11543-6
- 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 $6 \leq n \leq 84393$.References
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Bibliographic 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