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

DOI:
https://doi.org/10.1090/S0002-9939-2012-11543-6

Published electronically:
June 25, 2012

MathSciNet review:
2996953

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Abstract | References | Similar Articles | Additional Information

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 $6 \leq n \leq 84393$.

- Manjul Bhargava,
*The density of discriminants of quartic rings and fields*, Ann. of Math. (2)**162**(2005), no. 2, 1031–1063. MR**2183288**, DOI https://doi.org/10.4007/annals.2005.162.1031 - Manjul Bhargava,
*The density of discriminants of quintic rings and fields*, Ann. of Math. (2)**172**(2010), no. 3, 1559–1591. MR**2745272**, DOI https://doi.org/10.4007/annals.2010.172.1559 - Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier,
*A survey of discriminant counting*, Algorithmic number theory (Sydney, 2002) Lecture Notes in Comput. Sci., vol. 2369, Springer, Berlin, 2002, pp. 80–94. MR**2041075**, DOI https://doi.org/10.1007/3-540-45455-1_7 - H. Davenport and H. Heilbronn,
*On the density of discriminants of cubic fields. II*, Proc. Roy. Soc. London Ser. A**322**(1971), no. 1551, 405–420. MR**491593**, DOI https://doi.org/10.1098/rspa.1971.0075 - Jordan S. Ellenberg and Akshay Venkatesh,
*The number of extensions of a number field with fixed degree and bounded discriminant*, Ann. of Math. (2)**163**(2006), no. 2, 723–741. MR**2199231**, DOI https://doi.org/10.4007/annals.2006.163.723 - Gunter Malle,
*On the distribution of Galois groups*, J. Number Theory**92**(2002), no. 2, 315–329. MR**1884706**, DOI https://doi.org/10.1006/jnth.2001.2713 - J. Pila,
*Density of integer points on plane algebraic curves*, Internat. Math. Res. Notices**18**(1996), 903–912. MR**1420555**, DOI https://doi.org/10.1155/S1073792896000554 - Wolfgang M. Schmidt,
*Number fields of given degree and bounded discriminant*, Astérisque**228**(1995), 4, 189–195. Columbia University Number Theory Seminar (New York, 1992). MR**1330934** - David J. Wright,
*Distribution of discriminants of abelian extensions*, Proc. London Math. Soc. (3)**58**(1989), no. 1, 17–50. MR**969545**, DOI https://doi.org/10.1112/plms/s3-58.1.17

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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

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.