Class numbers, cyclic simple groups, and arithmetic

Finiteness of Class Numbers for Algebraic Groups

Let G be an affine group scheme of finite type over a global field F . (We do not assume G to be reductive or smooth or connected.) Let AF denote the locally compact adele ring of F , S be a finite non-empty set of places of F that contains the archimedean places, and AF the factor ring of adeles with vanishing component along S; for FS = ∏ v∈S Fv, we have AF = FS ×AF . Recall that if X is a fi...

Simple Current Actions of Cyclic Groups

Quadratic Fields with Cyclic 2-class Groups

For any integer k ≥ 1, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order 2. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class to be a square. In memory of David Hayes.

Dihedral and cyclic extensions with large class numbers

This paper is a continuation of [2]. We construct unconditionally several families of number fields with large class numbers. They are number fields whose Galois closures have as the Galois groups, dihedral groups Dn, n = 3, 4, 5, and cyclic groups Cn, n = 4, 5, 6. We first construct families of number fields with small regulators, and by using the strong Artin conjecture and applying some modi...

Relative K-groups and class field theory for arithmetic surfaces

Here CH0(X) denotes the Chow group of zero-cycles on X modulo rational equivalence and π̃ 1 (X) ab is the modified abelianized étale fundamental group, which classifies finite abelian étale coverings of X in which every real point splits completely (if X is defined over a totally imaginary number field, then this is just the usual abelianized étale fundamental group). By Lang [La], the homomorph...