Explicit Lower bounds for residues at $s=1$ of Dedekind zeta functions and relative class numbers of CM-fields

Abstract:

Let $S$ be a given set of positive rational primes. Assume that the value of the Dedekind zeta function $\zeta _K$ of a number field $K$ is less than or equal to zero at some real point $\beta$ in the range ${1\over 2} <\beta <1$. We give explicit lower bounds on the residue at $s=1$ of this Dedekind zeta function which depend on $\beta$, the absolute value $d_K$ of the discriminant of $K$ and the behavior in $K$ of the rational primes $p\in S$. Now, let $k$ be a real abelian number field and let $\beta$ be any real zero of the zeta function of $k$. We give an upper bound on the residue at $s=1$ of $\zeta _k$ which depends on $\beta$, $d_k$ and the behavior in $k$ of the rational primes $p\in S$. By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields $K$ which depend on the behavior in $K$ of the rational primes $p\in S$. We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields.