Computing Selmer groups associated to mod p Galois representations

TL;DR

This paper introduces explicit computational methods for Selmer groups linked to mod p Galois representations over number fields, exploring their ranks and periods to provide evidence for a mod p version of the Bloch-Kato conjecture.

Contribution

It develops and implements in Magma new algorithms to compute Selmer groups and compare their ranks with cohomological periods, supporting a conjectural link.

Findings

Evidence supporting a mod p Bloch-Kato conjecture

Explicit Magma implementations for Selmer group computations

Comparison of Selmer ranks with cohomological periods

Abstract

We present methods to compute Selmer groups associated to mod p Galois representations rho over a number field K, with a particular focus on comparing their ranks with periods coming from cohomology classes associated to rho by Serre's conjecture. This provides evidence for a loose version of a "mod p Bloch-Kato conjecture", where the vanishing of a period is predicted to capture the presence of rank in a Selmer group. Our methods are explicit, and implemented in Magma.