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

Eric Larson; Larry Rolen

paper

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

arXiv:1107.1182

Abstract

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 $N (n, A_{n}, X) \sim C_{n} X^{1/2} (lo g X)^{b_{n}}$ , for constants $b_{n}$ and $C_{n}$ . For $5 < n < 84394$ , the best known upper bound is $N (n, A_{n}, X) ≪ X^{\frac{n + 2}{4}}$ ; this bound follows from Schmidt's Theorem, which implies there are $≪ X^{\frac{n + 2}{4}}$ 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 $N (n, A_{n}, X) ≪ X^{\frac{n ^{2} - 2}{4 ( n - 1 )} + ε}$ , thereby improving the best previous exponent by approximately 1/4 for $5 < n < 84394$ .