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Dimension of the deformation space of ordinary representations in the cyclotomic limit

Ashay A. Burungale, Laurent Clozel, Barry Mazur

TL;DR

The paper investigates the dimension of the cyclotomic ordinary deformation space for two-dimensional Galois representations, showing that while formal smoothness can occur under standard global hypotheses, the number of deformation-parameters can be made arbitrarily large by introducing ramification. It develops a local-cohomology framework for the adjoint representation in the cyclotomic tower and proves that the limit global cohomology is free over the Iwasawa algebra, forcing a formal power-series structure for the limit deformation ring when Noetherian. It then provides an explicit $p=5$ example via level-raising that yields a family of deformation rings with unbounded dimension, and extends the results to general odd primes and weights $\ abla \, {\kappa}=2$ and $\kappa=p+1$, using Ribet primes and automorphic lifts. Overall, the work demonstrates that the cyclotomic ordinary deformation space can have arbitrarily large dimension, highlighting the nuanced interplay between local-global conditions and the Noetherianity of deformation rings in Iwasawa theory.

Abstract

The weight two ordinary deformations are unobstructed in the cyclotomic limit under certain assumptions. We show that such an ordinary deformation ring over the cyclotomic tower can have arbitrarily large dimension.

Dimension of the deformation space of ordinary representations in the cyclotomic limit

TL;DR

The paper investigates the dimension of the cyclotomic ordinary deformation space for two-dimensional Galois representations, showing that while formal smoothness can occur under standard global hypotheses, the number of deformation-parameters can be made arbitrarily large by introducing ramification. It develops a local-cohomology framework for the adjoint representation in the cyclotomic tower and proves that the limit global cohomology is free over the Iwasawa algebra, forcing a formal power-series structure for the limit deformation ring when Noetherian. It then provides an explicit example via level-raising that yields a family of deformation rings with unbounded dimension, and extends the results to general odd primes and weights and , using Ribet primes and automorphic lifts. Overall, the work demonstrates that the cyclotomic ordinary deformation space can have arbitrarily large dimension, highlighting the nuanced interplay between local-global conditions and the Noetherianity of deformation rings in Iwasawa theory.

Abstract

The weight two ordinary deformations are unobstructed in the cyclotomic limit under certain assumptions. We show that such an ordinary deformation ring over the cyclotomic tower can have arbitrarily large dimension.
Paper Structure (17 sections, 24 theorems, 45 equations)

This paper contains 17 sections, 24 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Ordinary deformations over cyclotomic tower
  3. Setup
  4. Residual representation
  5. Ordinary deformations
  6. Main result
  7. Local cohomology
  8. Local cohomology of the adjoint
  9. Main result
  10. An explicit example for $p=5$
  11. The example
  12. Setting
  13. Variant
  14. Growth of deformation rings
  15. General $\bar{\rho}$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Assume $R_\infty$ is Noetherian. Moreover, assume that Then it is formally smooth, i.e. for some $s \geq 1$ (cf. BC).

Theorems & Definitions (39)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Lemma 2.7
  • ...and 29 more