Paper detail

Reflection principles for class groups

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 $\mathbb{Z}/l\mathbb{Z}\rtimes\mathbb{Z}/r\mathbb{Z}$. The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.