No Riemannhurwitz formula for the pranks of relative class groups
Abstract
We disprove, by means of numerical examples, the existence of a RiemannHurwitz formula for the pranks of relative class groups in a pramified pextension K/k of number fields of CMtype containing ?\_p. In the cyclic case of degree p, under some assumptions on the pclass group of k, we prove some properties of the Galois structure of the pclass group of K; but we have found, through numerical experimentation, that some theoretical group structures do not exist in this particular situation, and we justify this fact. Then we show, in this context, that Kida's formula on lambda invariants is valid for the pranks if and only if the pclass group of K is reduced to the group of ambiguous classes, which is of course not always the case.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.07115
 Bibcode:
 2015arXiv151207115G
 Keywords:

 Mathematics  Number Theory
 EPrint:
 6 pages + tables num\'eriques