Densities of quartic fields with even Galois groups

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 $|D_K| \le X$. 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.