Upper bounds on number fields of given degree and bounded discriminant (Q2093537)

The authors give an upper bound for the number \(N_n(X)\) of number fields of degree \(n\) with discriminant bounded by \(X\): \(N_n(X) =O(X^{c(\log n)^2})\) for an explicit constant \(c\). The authors improve previous results of many authors. Their strategy follows the ideas of \textit{J. S. Ellenberg} and \textit{A. Venkatesh} [Ann. Math. (2) 163, No. 2, 723--741 (2006; Zbl 1099.11068)], and of \textit{J.-M. Couveignes} [Ann. Math. (2) 192, No. 2, 487--497 (2020; Zbl 1454.11190)].