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Densities of quartic fields with even Galois groups

Authors:	Siman Wong

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

Abstract:	Let be the number of degree number fields with Galois group and whose discriminant satisfies . 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 whose Galois groups are smaller than , confirming a question of Gallagher. Unconditionally we have $N(4, A_4, X) ll_epsilon X^{5/6 + epsilon}$ , and that the -class groups of almost all Abelian cubic fields have size $ll_epsilon D_k^{1/3+epsilon}$ . The proofs depend on counting integral points on elliptic fibrations.

Keywords:	Class groups discriminants elliptic curves elliptic fibrations Galois groups integral points quartic fields

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