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Computing Selmer groups associated to mod p Galois representations

Lewis Combes

TL;DR

The work delivers explicit, Magma-implemented algorithms to compute mod p Selmer groups for two-dimensional Galois representations over number fields, focusing on relaxed, nearly-ordinary, and unramified variants. It translates cohomological data into abelian extensions via class field theory and ray class fields, enabling finite, computable Selmer groups. Through concrete examples and statistics (notably GL2(F2) and SD16 in GL2(F3)) and a Serre-based period framework, the paper provides computational evidence for a mod p analogue of Bloch-Kato-type relationships, particularly in the nearly-ordinary case. These results offer a practical approach to probing the interplay between periods and Selmer ranks in the mod p setting and lay groundwork for further refinements of mod p conjectures.

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.

Computing Selmer groups associated to mod p Galois representations

TL;DR

The work delivers explicit, Magma-implemented algorithms to compute mod p Selmer groups for two-dimensional Galois representations over number fields, focusing on relaxed, nearly-ordinary, and unramified variants. It translates cohomological data into abelian extensions via class field theory and ray class fields, enabling finite, computable Selmer groups. Through concrete examples and statistics (notably GL2(F2) and SD16 in GL2(F3)) and a Serre-based period framework, the paper provides computational evidence for a mod p analogue of Bloch-Kato-type relationships, particularly in the nearly-ordinary case. These results offer a practical approach to probing the interplay between periods and Selmer ranks in the mod p setting and lay groundwork for further refinements of mod p conjectures.

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.

Paper Structure

This paper contains 29 sections, 6 theorems, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Structure of the paper
  4. Acknowledgments
  5. Preliminaries and notation
  6. Galois cohomology and Selmer groups
  7. Cohomology to class field theory
  8. Inflation-restriction
  9. Homomorphisms to number fields
  10. Translating local conditions
  11. The relaxed condition
  12. The unramified condition
  13. The nearly-ordinary condition
  14. Class field theory
  15. Technical considerations
  16. ...and 14 more sections

Key Result

Theorem 4.3

Let $V$ be as above. Then non-trivial lines in $\mathrm{Hom}_{\mathrm{Gal}(L/K)}(G_L,V)$ are in bijection with extensions $M/L$ such that

Figures (2)

  • Figure 1: Average ranks of the relaxed, nearly-ordinary and unramified mod $2$ Selmer groups over various number fields.
  • Figure 2: Average ranks of the relaxed and unramified Selmer groups of nearly-ordinary $\mathrm{SD}_{16}$ representations over $\mathbb{Q}$, and ranks of the nearly-ordinary Selmer groups associated to ramified and unramified quotients.

Theorems & Definitions (22)

  • Definition 2.1
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 4.1
  • Remark 4.2
  • Theorem 4.3
  • ...and 12 more