Bart de Smit

Class Number Relations from a Computational Point of View

Brauer and Kuroda showed in the fifties how in a Galois extension of number fields, relations between permutation characters of subgroups provide relations between invariants, such as the discriminant, class number and regulator, of the corresponding intermediate fields. In this paper we investigate various computational aspects of these relations, we present examples, and we give a method to automatically produce class number formulas.

On Arithmetically Equivalent Number Fields of Small Degree

For each integer n, let Sn be the set of all class number quotients h(K)/h(K?) for number fields K and K? of degree n with the same zeta-function.In this note we will give some explicit results on the finite sets Sn, for small n. For example, for every x ? Sn with n ? 15, x or x-1 is an integer that is a prime power dividing 214 ? 36 ? 53.

Generating Arithmetically Equivalent Number Fields with Elliptic Curves

In this note we address the question whether for a given prime number p, the zeta-function of a number field always determines the p-part of its class number. The answer is known to be no for p = 2. Using torsion points on elliptic curves we give for each odd prime p an explicit family of pairs of non-isomorphic number fields of degree 2p + 2 which have the same zeta-function and which satisfy a necessary condition for the fields to have distinct p-class numbers. By computing class numbers of fields in this family for p=3 we find examples of fields with the same zeta-function whose class numbers differ by a factor 3.