Reflection principles for class groups

2016/05/14 by Klys, Jack
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1605.04371

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 ℙ¹/\mathbbF_p. This proves a conjecture of Lemmermeyer 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, D_2l and ℤ/lℤ\rtimesℤ/rℤ. The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.

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