Probabilistic properties of number fields
Given a number field \(K\), let \(n(K):= [K:\mathbb{Q}]\), let \(d_K\) denote the absolute value of the discriminant of \(K\), let \(\delta_K:=\log\log d_K\), let \(N(K)\) be the smallest rational prime that does not split completely in \(K\), let \(G(K)\) be the Galois group of the normal closure of \(K\) over \(\mathbb{Q}\), and let \[ F({\mathfrak G},X):= \{K\mid G(K)={\mathfrak G},\,d_K\leq X\}. \] Write \(L(s,\chi_K):= \zeta_K(s)\zeta_{\mathbb{Q}}(s)^{-1}\), where \(\zeta_K(s)\) is the Dedekind zeta-function of a number field \(K\) and \(L(s,\chi_K)\) is the corresponding Artin \(L\)-function. Assume that \[ X^{a({\mathfrak G})}\ll_{\mathfrak G}\,\text{card\,}F({\mathfrak G}, X)\ll_{{\mathfrak G},\varepsilon} X^{a({\mathfrak G})+\varepsilon},\;\varepsilon> 0, \] for some \(a({\mathfrak G})>0\) (``Malle's conjecture''), that the Artin \(L\)-functions \(L(s,\chi_K)\) are \(\mathrm{GL}(n(K)- 1,\mathbb{Q})\)-automorphic \(L\)-functions in the sense of Langlands (``strong Artin conjecture''), and a somewhat technical (in general, unproved) hypothesis on certain isobaric representations of the groups \(\mathrm{GL}(l,\mathbb{Q})\), the authors prove that the following estimates \[ \delta^{-1}_K\ll_{\mathfrak G} L(1,\chi_K)\ll_{\mathfrak G} \delta^{n-1}_K,\;N(K)\ll_{\mathfrak G}(\log d_K)^{\alpha({\mathfrak G})}, \] and \(-(n-1)\alpha_1({\mathfrak G}) \delta_K+ O_{\mathfrak G}(1)\leq {L'\over L} (1,\chi_K)\leq \alpha_1({\mathfrak G}) \delta_K+ O_{\mathfrak G}(1)\), with some positive \(\alpha({\mathfrak G})\), \(\alpha_1({\mathfrak G})\), hold for all but \(O(X^\gamma)\) fields \(K\) in \(F({\mathfrak G},X)\) whenever \(\gamma> 0.99\) \(a({\mathfrak G})\). Since all the authors' assumptions have been proved to hold for \[ {\mathfrak G}\in \{C_2, S_3, A_4, S_4, C_p, D_p\mid p\in P\}, \] where \(P\) stands for the set of the odd rational primes, their estimates hold unconditionally in this case.
scientific article; zbMATH DE number 6341853
scientific article