Computing Selmer groups associated to mod p Galois representations

2025/12/15 by Lewis Combes, Combes, Lewis
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2512.12899

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.

Citations

• An algorithm to compute Selmer groups via resolutions by permutations modules
• A torsion Jacquet--Langlands correspondence
• Modular forms and period polynomials

Related

• Deformations of reducible Galois representations with large Selmer p-rank
• An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod-p cohomology of GL(n,Z)
• Selmer groups as flat cohomology groups
• On the Bloch-Kato conjecture for some four-dimensional symplectic Galois representations
• Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity