UPPER BOUNDS FOR THE NUMBER OF NUMBER FIELDS WITH ALTERNATING GALOIS GROUP

Let N(n, A(n), X) be the number 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) similar to C-n . X-1/2 . (log X)(bn) for constants b(n) and C-n. For 6 <= n <= 84393, the best known upper bound is N(n, A(n), X) << Xn+2/4, by Schmidt's theorem, which implies there are << Xn+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) << Xn2-2/4(n-1) + epsilon, thereby improving the best previous exponent by approximately 1/4 for 6 <= n <= 84393.