Upper bounds for the number of number fields with alternating Galois group. (English) Zbl 1296.11145
Summary: 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 G. Malle [J. Number Theory 92, 315–322 (2002; Zbl 1022.11058), we expect that
\[
N(n, A_n, X)\sim C_n\cdot X^{\frac {1}{2}} \cdot (\log X)^{b_n}
\]
for constants \( b_n\) and \( C_n\). For \( 6 \leq n \leq 84393\), the best known upper bound is
\[
N(n, A_n, X) \ll X^{\frac {n + 2}{4}},
\]
by Schmidt’s theorem [W. M. Schmidt [Columbia University number theory seminar, New York, 1992. Astérisque 228, 189–195 (1995; Zbl 0827.11069)], which implies there are \( \ll X^{\frac {n + 2}{4}}\) number fields of degree \(n\). (For \( n > 84393\), there are better bounds due to J. S. Ellenberg and A. Venkatesh [Ann. Math. (2) 163, No. 2, 723–741 (2006; Zbl 1099.11068)].) We show, using the important work of J. Pila [Int. Math. Res. Not. 1996, No. 18, 903–912 (1996; Zbl 0973.11085)] on counting integral points on curves, that
\[
N(n, A_n, X) \ll X^{\frac {n^2 - 2}{4(n - 1)}+\varepsilon },
\]
thereby improving the best previous exponent by approximately \( \frac {1}{4}\) for \( 6 \leq n \leq 84393\).
References:
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