On unit signatures and narrow class groups of odd degree abelian number fields

Abstract

For an abelian number field of odd degree, we study the structure of its 2-Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow class groups in families where the degree and Galois group are fixed. Contents 1. Introduction 2. Properties of the 2-Selmer group and its signature spaces 3. Galois module structure for invariants of odd Galois number fields 4. Structural results 5. Conjectures 6. Computations Appendix A. Cyclic cubic fields with signature rank 1 (with Noam Elkies) References