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

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

2011-07-06

arxiv.org/abs/1107.1182v1

Mathematics

Number Theory

Scientific paper

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 $N(n, A_n, X) \sim C_n X^{1/2} (\log X)^{b_n}$, for constants $b_n$ and $C_n$. 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 $5 < n < 84394$.

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