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The Taylor-Wiles method for reductive groups

Dmitri Whitmore

Abstract

We construct a local deformation problem for residual Galois representations $\barρ$ valued in an arbitrary reductive group $\hat{G}$ which we use to develop a variant of the Taylor-Wiles method. Our generalization allows Taylor-Wiles places for which the image of Frobenius is semisimple, a weakening of the regular semisimple constraint imposed previously in the literature. We introduce the notion of $\hat{G}$-adequate subgroup, our corresponding 'big image' condition. When $\hat{G}$ is a simply connected simple group of type $\mathrm{C}$ or of exceptional type and $\hat{G} \to \mathrm{GL}_n$ is a faithful irreducible representation of minimal dimension, we show that a subgroup is $\hat{G}$-adequate if it is $\mathrm{GL}_n$-irreducible and the residue characteristic is sufficiently large. We apply our ideas to the case $\hat{G} = \mathrm{GSp}_4$ and prove a modularity lifting theorem for abelian surfaces over a totally real field $F$ which holds under weaker hypotheses than in the work of Boxer-Calegari-Gee-Pilloni. We deduce some modularity results for elliptic curves over quadratic extensions of $F$.

The Taylor-Wiles method for reductive groups

Abstract

We construct a local deformation problem for residual Galois representations valued in an arbitrary reductive group which we use to develop a variant of the Taylor-Wiles method. Our generalization allows Taylor-Wiles places for which the image of Frobenius is semisimple, a weakening of the regular semisimple constraint imposed previously in the literature. We introduce the notion of -adequate subgroup, our corresponding 'big image' condition. When is a simply connected simple group of type or of exceptional type and is a faithful irreducible representation of minimal dimension, we show that a subgroup is -adequate if it is -irreducible and the residue characteristic is sufficiently large. We apply our ideas to the case and prove a modularity lifting theorem for abelian surfaces over a totally real field which holds under weaker hypotheses than in the work of Boxer-Calegari-Gee-Pilloni. We deduce some modularity results for elliptic curves over quadratic extensions of .
Paper Structure (36 sections, 89 theorems, 188 equations, 5 tables)

This paper contains 36 sections, 89 theorems, 188 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Main ideas and theorems
  3. Structure of the paper
  4. Acknowledgements
  5. Setup for a local deformation problem
  6. Lifting the Lie algebra
  7. Lifting $M_{\bar{g}}$
  8. Functoriality properties of $M_{g}$
  9. A construction of $M_g^\circ$ over $\overline{\mathbb{Q}_p}$
  10. Deformation theory
  11. Deformation theory of representations
  12. Taylor--Wiles places
  13. Dual groups and the $\Delta_Q$-action
  14. Presentations of framed deformation rings in the number fields case
  15. $\hat{G}$-adequate subgroups
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $A$ be a complete Noetherian local $\mathbb{Z}_p$-algebra of residue field $\mathbb{F}_p$ and let $g \in \hat{G}(A)$ be a lift of $\bar{g}$. Then there is a canonical lift of the centralizer $M_{\bar{g}} = Z_{\hat{G}_{\mathbb{F}_p}}(\bar{g})$ to a closed reductive subgroup scheme $M_g$ of $\hat{

Theorems & Definitions (195)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 185 more