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The spin of prime ideals and level-raising of even Galois representations

Marius Fischer, Peter Vang Uttenthal

TL;DR

The paper proves, under a short-cubical character-sum conjecture (C12), that primes p in the set C (p ≡ 1 mod 3, unramified in a fixed A4-extension, with Frobenius of order 3) raising the level of a given even Galois representation have density 2/3 in C, as conjectured by Ramakrishna. It introduces a spin symbol attached to degree-1 primes over Q in a non-Galois setting and links level-raising to a cubic-residue criterion captured by this spin, using class-field theory and Artin reciprocity. The argument combines a Vinogradov-type sieve with delicate lattice-point counting and a fundamental-domain reduction to manage non-Galois complications, culminating in a density result and a Selmer-rank corollary. This work represents the first application of spin methods to deformation-theoretic questions for Galois representations and suggests broader applicability to similar level-raising problems.

Abstract

By extending the notion of spin of prime ideals, we show that a short character sum conjecture implies that the set of primes raising the level of a certain even Galois representation has density 2/3, as conjectured by Ramakrishna in 1998.

The spin of prime ideals and level-raising of even Galois representations

TL;DR

The paper proves, under a short-cubical character-sum conjecture (C12), that primes p in the set C (p ≡ 1 mod 3, unramified in a fixed A4-extension, with Frobenius of order 3) raising the level of a given even Galois representation have density 2/3 in C, as conjectured by Ramakrishna. It introduces a spin symbol attached to degree-1 primes over Q in a non-Galois setting and links level-raising to a cubic-residue criterion captured by this spin, using class-field theory and Artin reciprocity. The argument combines a Vinogradov-type sieve with delicate lattice-point counting and a fundamental-domain reduction to manage non-Galois complications, culminating in a density result and a Selmer-rank corollary. This work represents the first application of spin methods to deformation-theoretic questions for Galois representations and suggests broader applicability to similar level-raising problems.

Abstract

By extending the notion of spin of prime ideals, we show that a short character sum conjecture implies that the set of primes raising the level of a certain even Galois representation has density 2/3, as conjectured by Ramakrishna in 1998.

Paper Structure

This paper contains 12 sections, 30 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Notation
  4. The cubic residue symbol
  5. The spin symbol
  6. Short character sums
  7. Vinogradov's sieve
  8. A fundamental domain
  9. Counting ideals of squarefull norm
  10. Sums of type I
  11. Sums of type II
  12. Level-raising of even Galois representations

Key Result

Theorem 1.1

Assume Conjecture $C_{12}$. Then the set of primes raising the level of $\rho$ has density $2/3$ in $\mathcal{C}$, i.e.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 4.1
  • proof
  • ...and 45 more