The strong Artin conjecture and large class numbers (Q2922411)

In this article, the behavior of the class number \(h_K\) of a number field is studied as \(K\) runs through the family \({\mathfrak K}(n,G,r_1,r_2)\) of number fields with fixed degree \(n = r_1 + 2r_2\) and signature \((r_1,r_2)\) whose normal closure has Galois group \(G\). The class number formula provides us with an upper bound for \(h_K\) if we can find upper bounds for \(L(1,\rho) = \zeta_K(s)/\zeta(s)\) (such as the one given by \textit{W. Duke} [Compos. Math. 136, No. 1, 103--115 (2003; Zbl 1013.11072)]) and lower bounds for the regulator (for example those provided by \textit{J. H. Silverman} [J. Number Theory 19, 437--442 (1984; Zbl 0552.12003)]). For totally real number fields, these upper bounds have the form \(h_K \ll \sqrt{d_K} (\log \log d_K/\log d_K)^{n-1}\). Duke asked whether this upper bound is best possible and showed hat the answer is positive if \(L(s,\rho)\) is entire and satisfies the GRH. The special cases \(n=2\) and \(n = 3\) were proved unconditionally by \textit{J. E. Littlewood} [Proc. Lond. Math. Soc. (2) 27, 358--372 (1928; JFM 54.0206.02)] and \textit{R. C. Daileda} [Acta Arith. 125, No. 3, 215--255 (2006; Zbl 1158.11044)], respectively. In this article, the author proves Dukes result unconditionally for \(n = 4\) by showing that the assumptions in Duke's theorem may be replaced by the assumption that \(L(s,\rho)\) satisfies the strong Artin conjecture (that the Artin \(L\)-functions are cuspidal automorphic \(L\)-functions of \(\mathrm{GL}(\ell)\) over \(\mathbb Q\)), and proving this conjecture for 3-dimensional Galois representations \(\rho\) of \(S_4\) or \(A_4\).