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A Local-Global Study of Obstructed Deformation Problems -- I

Bartu Bingol

TL;DR

The paper develops a local-global framework for obstructed deformations of two-dimensional residual Galois representations attached to weight $2$ newforms, using Poitou–Tate duality to separate local and global obstructions. It analyzes level-raising as a source of a single local obstruction at a new prime $q$, providing an explicit Brauer-group cocycle and a one-relator deformation-ring description in Steinberg-type settings. It then treats global obstructions via strict congruence primes and Selmer groups $\Sh^1$, giving criteria to detect when obstructions are genuinely global and how they relate to non-optimal levels; practical methods for identifying strict congruence primes are discussed and explicit deformation rings are given in notable cases. Overall, the work clarifies how local level-raising phenomena interact with global congruence obstructions to shape the deformation theory of modular Galois representations, with concrete consequences for the structure of universal deformation rings and level optimization.

Abstract

We study obstructed deformation problems for two-dimensional residual Galois representations arising from weight~$2$ newforms of level~$N$. Using Poitou-Tate duality, we isolate local and global sources of obstructions and give concrete criteria for when they occur. In several cases we also describe the resulting universal deformation ring explicitly.

A Local-Global Study of Obstructed Deformation Problems -- I

TL;DR

The paper develops a local-global framework for obstructed deformations of two-dimensional residual Galois representations attached to weight newforms, using Poitou–Tate duality to separate local and global obstructions. It analyzes level-raising as a source of a single local obstruction at a new prime , providing an explicit Brauer-group cocycle and a one-relator deformation-ring description in Steinberg-type settings. It then treats global obstructions via strict congruence primes and Selmer groups , giving criteria to detect when obstructions are genuinely global and how they relate to non-optimal levels; practical methods for identifying strict congruence primes are discussed and explicit deformation rings are given in notable cases. Overall, the work clarifies how local level-raising phenomena interact with global congruence obstructions to shape the deformation theory of modular Galois representations, with concrete consequences for the structure of universal deformation rings and level optimization.

Abstract

We study obstructed deformation problems for two-dimensional residual Galois representations arising from weight~ newforms of level~. Using Poitou-Tate duality, we isolate local and global sources of obstructions and give concrete criteria for when they occur. In several cases we also describe the resulting universal deformation ring explicitly.
Paper Structure (24 sections, 11 theorems, 63 equations)

This paper contains 24 sections, 11 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Notation and standing assumptions
  3. Deformations and deformation rings
  4. Outline and main results
  5. Deformation theory of two-dimensional Galois representations
  6. The deformation functor and its tangent space
  7. Obstructions
  8. Obstructions arising from level raising
  9. Global setup and the Poitou--Tate reduction
  10. Tame generators at $q$ and the invariant line
  11. Pairings and the obstruction detected by an exponent
  12. The diagram and the explicit recipe in the Brauer group
  13. Consequences for the obstruction space and the deformation ring
  14. Global obstructions and strict congruence primes
  15. Notation and standing hypotheses
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $\bar{\rho}:G_{\mathbb{Q},S}\to \mathrm{GL}_2(\mathbb{F}_\ell)$ arise from a weight $2$ newform of level $N$, where $S$ contains the primes dividing $N\ell$, and assume $\ell>3$. Suppose the deformation problem for $\bar{\rho}$ over $G_{\mathbb{Q},S}$ is unobstructed. Let $q\notin S$ be a prime

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Definition 2.3
  • Lemma 3.2
  • proof
  • Remark 3.3: Local level-raising vs. non-optimal level
  • Proposition 4.1: Poitou--Tate reduction to Sha1
  • ...and 14 more