TAMELY RAMIFIED TOWERS AND DISCRIMINANT BOUNDS FOR NUMBER FIELDS FARSHID HAJIR AND CHRISTIAN MAIRE Abstract. The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R 2m be the minimal root discriminant for totally complex number fields of degree 2m, and put α 0 = lim inf m R 2m . One knows that α 0 ≥ 4πe γ ≈ 22.3, and, assuming the Generalized Riemann Hypothesis, α 0 ≥ 8πe γ ≈ 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that α 0 < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, 1) we allow tame ramification in the tower, and 2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate α 0 < 83.9. 1. Introduction Suppose K is a number field of degree n = r 1 +2r 2 and signature (r 1 ,r 2 ), i.e. K has r 1 real, and r 2 pairs of complex conjugate, embeddings. The root discriminant rd K of K is defined by rd K = |d K | 1/n where d K is the discriminant of K. This invariant may be thought of as measuring the density of the integer lattice O K embedded in K ⊗ R ∼ = R r1 × C r2 in the standard way (under the trace norm). In 1891, Minkowski [Mi] used his “geometry of numbers” to give an explicit lower bound for the discriminant of K which is exponential in the degree of K. Minkowski’s estimate was substantially improved over the years using refinements of his technique. In 1974, Stark [St] introduced an analytic method (based on a study of the zeros of the Dedekind zeta function) for proving discriminant lower bounds. Stark’s approach was refined by Odlyzko [O1], [O2] to improve substantially the estimates obtained from geometry of numbers. Meanwhile, Serre [Se1] introduced a variation based on the Guinand-Weil explicit formulas which was further investigated by Odlyzko [O3] and Poitou [P], leading to the best known lower bounds today (see the survey paper of Odlyzko [O4] for more details). Naturally, the best bounds are obtained under the assumption of the Generalized Riemann Hypothesis (GRH). The asymptotic version of these bounds is as follows: If the degree n of K tends to infinity, then rd K ≥ A 2r2/n B r1/n e o(n) ; (1) here, unconditionally, we can take A =4πe γ ≈ 22.3,B =4πe 1+γ ≈ 60.8, and on GRH, A =8πe γ ≈ 44.7,B =8πe γ+π/2 ≈ 215.3, where γ =0.577 ··· is Euler’s constant (see Odlyzko [O4]). Lower bounds for discriminants have been found to be very useful in a wide variety of applications, the main one being estimation of class numbers (e.g. Masley [Mas] and Yamamura [Y2]; many more references can be found in [O4, References D]). For 1