Upper bounds of root discriminant lower bounds (Q494959)

The main goal of the paper is to study the existence of families of number fields with small root discriminant \(rd_K=|d_K|^{1/n_K}, n_K=[K:\mathbb{Q}]\) and additional restrictions on the number of real places \(r_1(K)\). Under the generalized Riemann hypothesis, the author proves that, fixing a fraction of the form \(a/(3^b m)\in [0,1]\cup \mathbb{Q}, 3\nmid m,\) there exists an infinite sequence of Galois extensions \(K_1\subset K_2\subset \dots,\) such that \(r_1(K_i)/n_{K_i}=a/(3^bm)\) for all \(i\) and \[ rd_{K_i}\leq 19.59316 + \frac{m-1}{m}(2\log m+2\log\log m+6.813445)+O\left(\frac{\log n_{K_i}+\log m}{m n_{K_i}}\right). \] In order to get this bound, the author uses the GRH form of the effective Chebotarev density theorem applied to the narrow class field of an explicit infinite 3-class field tower of a real quadratic field. The last step consists of composing it with a field having an appropriate number of real and complex places. The unconditional results are much weaker. The sequence of fields \(K_i\) satisfying \(r_1(K_i)/n_{K_i}=t\) and \(\log rd_{K_i} \leq c n_{K_i} \log n_{K_i}\) is constructed via an explicit family of polynomials splitting completely over \(\mathbb{Z}\).