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On flat even deformation rings

Peter Vang Uttenthal

TL;DR

This paper develops parity-free lifting techniques for Galois representations by employing the Galois cohomological method for reductive groups to study balanced global deformation settings. It proves the existence of finite flat global even deformation rings $R_d \simeq \mathbb{Z}_p[[T]]/(T^d+\cdots)$ in rank-one balanced cases, even with nontrivial dual Selmer groups, and provides explicit $p=3$ examples built from $A_4$-extensions to illustrate flatness and the Weierstrass framework. A key advancement is showing that, under suitable auxiliary primes and Leopoldt-type hypotheses for attached number fields, flatness can be achieved at the minimal level, linking Selmer balance with arithmetic conjectures. The results extend parity-free lifting methods beyond odd representations and offer concrete instances and conditional criteria for flat even deformation rings, with implications for characteristic-zero lifts and level-raising phenomena.

Abstract

In the presence of a nontrivial dual Selmer group, certain global even deformation rings are shown to be finite and flat over $\mathbb{Z}_p$. Previously, flatness was only known in established cases of Langlands reciprocity in the odd parity. By techniques from global class field theory, explicit examples of even representations are computed to which the results apply. For even representations $\overlineρ$ in an explicit family, it is observed that if Leopoldt's conjecture is true for a certain number field attached to $\overlineρ$, then the global even deformation ring is flat at the minimal level.

On flat even deformation rings

TL;DR

This paper develops parity-free lifting techniques for Galois representations by employing the Galois cohomological method for reductive groups to study balanced global deformation settings. It proves the existence of finite flat global even deformation rings in rank-one balanced cases, even with nontrivial dual Selmer groups, and provides explicit examples built from -extensions to illustrate flatness and the Weierstrass framework. A key advancement is showing that, under suitable auxiliary primes and Leopoldt-type hypotheses for attached number fields, flatness can be achieved at the minimal level, linking Selmer balance with arithmetic conjectures. The results extend parity-free lifting methods beyond odd representations and offer concrete instances and conditional criteria for flat even deformation rings, with implications for characteristic-zero lifts and level-raising phenomena.

Abstract

In the presence of a nontrivial dual Selmer group, certain global even deformation rings are shown to be finite and flat over . Previously, flatness was only known in established cases of Langlands reciprocity in the odd parity. By techniques from global class field theory, explicit examples of even representations are computed to which the results apply. For even representations in an explicit family, it is observed that if Leopoldt's conjecture is true for a certain number field attached to , then the global even deformation ring is flat at the minimal level.
Paper Structure (11 sections, 17 theorems, 99 equations)

This paper contains 11 sections, 17 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Summary
  4. Acknowledgments
  5. The Galois cohomological method for reductive groups
  6. Balanced global settings of rank one
  7. Auxiliary primes for $p\geq 5$
  8. Auxiliary primes for $p=3$
  9. Existence of $p$-adic representations without annihilating dual Selmer
  10. Explicit examples of even deformation rings that are flat over $\mathbb{Z}_3$
  11. Leopoldt's conjecture and flatness of even deformation rings

Key Result

Theorem 1

Let $S$ be a finite set of places in $\mathbb{Q}$ containing the prime $p$ and the infinite place. Let $\mathbb{Q}_S$ be the maximal extension of $\mathbb{Q}$ unramified outside $S$ and consider an irreducible even representation Suppose the global setting is balanced of rank one, in the sense that the Selmer and dual Selmer groups are of equal rank one: Let $d\geqslant 2$ be any integer. There

Theorems & Definitions (41)

  • Theorem 1
  • Corollary 2
  • Remark 3: concerning Theorem \ref{['petervang']}
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 7: Wiles' formula
  • proof
  • Lemma 8
  • proof
  • ...and 31 more