Proceedings of the American Mathematical Society

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



Densities of quartic fields with even Galois groups

Author: Siman Wong
Journal: Proc. Amer. Math. Soc. 133 (2005), 2873-2881
MSC (2000): Primary 11G05; Secondary 11G35, 11R16, 11R29
Published electronically: April 20, 2005
MathSciNet review: 2159764
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Abstract: Let $ N(d, G, X) $be the number of degree $d$ number fields $K$ with Galois group $G$ and whose discriminant $D_K$ satisfies $\vert D_K\vert \le X$. Under standard conjectures in diophantine geometry, we show that $ N(4, A_4, X) \ll_\epsilon X^{2/3+\epsilon} $, and that there are $ \ll_\epsilon N^{3+\epsilon} $monic, quartic polynomials with integral coefficients of height $\le N$whose Galois groups are smaller than $S_4$, confirming a question of Gallagher. Unconditionally we have $ N(4, A_4, X) \ll_\epsilon X^{5/6 + \epsilon} $, and that the $2$-class groups of almost all Abelian cubic fields $k$ have size $ \ll_\epsilon D_k^{1/3+\epsilon} $. The proofs depend on counting integral points on elliptic fibrations.

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Additional Information

Siman Wong
Affiliation: Department of Mathematics & Statistics, University of Massachusetts, Amherst, Massachusetts 01003-9305

Keywords: Class groups, discriminants, elliptic curves, elliptic fibrations, Galois groups, integral points, quartic fields
Received by editor(s): March 11, 2004
Received by editor(s) in revised form: June 7, 2004
Published electronically: April 20, 2005
Additional Notes: The author was supported in part by NSA grant H98230-05-1-0069
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2005 American Mathematical Society