Reflection principles for class groups

Abstract

We present several new examples of reflection principles which apply to both class groups of number fields and picard groups of of curves over ${\mathbb{P}}^1/{\mathbb{F}}_p$. This proves a conjecture of Lemmermeyer about equality of 2-rank in subfields of $A_4$, up to a constant not depending on the discriminant in the number field case, and exactly in the function field case. More generally we prove similar relations for subfields of a Galois extension with group $G$ for the cases when $G$ is $S_3$, $S_4$, $A_4$, $D_{2l}$ and $\mathbb{Z}/l\mathbb{Z}⋊\mathbb{Z}/r\mathbb{Z}$. The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.

22 pages