Upper Bounds for the Number of Number Fields with Alternating Galois Group

We study the number $N(n,A_n,X)$ 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 $N(n, A_n, X) \sim C_n X^{1/2} (\log X)^{b_n}$,forconstants$b_n$and$C_n$\mathrm{.}For$5 < n < 84394$, the best known upper bound is $N(n,A_n,X)\ll X^{\frac{n+2}{4}}$; this bound follows from 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 1/4 for $5<n<84394$.