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

We study the number N(n,An, X) of number fields of degree n whose Galois closure has Galois group An and whose discriminant is bounded by X. By a conjecture of Malle, we expect that N(n,An, X) ∼ Cn ·X 1 2 ·(logX)bn for constants bn and Cn. For 6 ≤ n ≤ 84393, the best known upper bound is N(n,An, X) � X n+2 4 ; this bound follows from Schmidt’s Theorem, which implies there are � X 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,An, X) � X n2−2 4(n−1), thereby improving the best previous exponent by approximately 4 for 6 ≤ n ≤ 84393.