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A classification of even representations onto 3-adic SL(2)

Peter Vang Uttenthal

TL;DR

The paper addresses the problem of classifying even tetrahedral-type Galois representations onto $\mathrm{SL}(2,\mathbb{Z}_3)$ with prime conductor. It employs a two-stage lifting strategy via global class field theory and idele twists, framed by a balanced Selmer-theoretic deformation setup, to construct an explicit series $\{\rho^{(\ell)}\}$ indexed by primes in $\Lambda^{(A_4)}$. Existence of the series is proved for primes satisfying detailed class-field-theory conditions, and exhaustiveness is established: any eligible representation arises from this series. The work also identifies explicit small primes (e.g., 163, 277, 349) producing the smallest surjective $3$-adic tetrahedral lifts and conjectures about infinitude and density, linking to Cohen–Lenstra heuristics for class groups. Overall, it provides a computable, principled classification of $3$-adic tetrahedral Galois representations with controlled ramification, grounded in deformation theory and global class field theory.

Abstract

This paper gives a classification of even representations onto $\operatorname{SL}(2,\mathbb{Z}_3)$ of prime conductor. In addition, an explicit algorithm based on global class field theory is exhibited, computing an exhaustive series of such even representations.

A classification of even representations onto 3-adic SL(2)

TL;DR

The paper addresses the problem of classifying even tetrahedral-type Galois representations onto with prime conductor. It employs a two-stage lifting strategy via global class field theory and idele twists, framed by a balanced Selmer-theoretic deformation setup, to construct an explicit series indexed by primes in . Existence of the series is proved for primes satisfying detailed class-field-theory conditions, and exhaustiveness is established: any eligible representation arises from this series. The work also identifies explicit small primes (e.g., 163, 277, 349) producing the smallest surjective -adic tetrahedral lifts and conjectures about infinitude and density, linking to Cohen–Lenstra heuristics for class groups. Overall, it provides a computable, principled classification of -adic tetrahedral Galois representations with controlled ramification, grounded in deformation theory and global class field theory.

Abstract

This paper gives a classification of even representations onto of prime conductor. In addition, an explicit algorithm based on global class field theory is exhibited, computing an exhaustive series of such even representations.
Paper Structure (9 sections, 17 theorems, 81 equations)

This paper contains 9 sections, 17 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Balancing the global setting
  4. The Selmer condition at the tame prime
  5. Balancing $\mathcal{N}_3$ and $\mathcal{N}_\infty$
  6. Classification
  7. Preliminary lemmas
  8. Existence of the series of even representations
  9. Proof that the series is exhaustive

Key Result

Theorem 1

The even representations onto $\operatorname{SL}(2,\mathbb{Z}_3)$ of prime conductors are indexed by a set of primes $\Lambda^{(A_4)}$ that can be computed by the following algorithm (straightforward to implement in magma or pari-gp): Loop through all primes $\ell$, and identify those $\ell$ satisfy Define $\Lambda^{(A_4)}$ to be the set of primes $\ell$ satisfying all of the four conditions liste

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Remark 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 25 more