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 curves over P1/Fp. This proves a conjecture of Lemmermeyer [3] about equality of 2-rank in subfields of A4, 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 S3, S4, A4, D2l and Z/lZ⋊Z/rZ. The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.