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Level-raising of even representations of tetrahedral type and equidistribution of lines in the projective plane

Peter Vang Uttenthal

TL;DR

The paper analyzes how often primes can be added to the level of even 3-adic tetrahedral-type Galois representations by linking local ramification data to global Selmer conditions via a balanced deformation framework. It provides extensive computational data for p=3 up to v ≤ 10^8 and proves a general density theorem: if the ramified lines in the local cohomology space are equidistributed in the projective plane, level-raising primes occur with density (p−1)/p, matching observed 2/3 for p=3. A central conjecture with Ramakrishna posits uniform distribution of these lines, connecting local probabilistic behavior to global deformation-theoretic phenomena. The work also discusses the rarity of even representations and the role of Selmer conditions at nice primes, highlighting the interplay between arithmetic geometry, Galois cohomology, and prime distribution in this setting.

Abstract

The distribution of primes raising the level of even Galois representations of tetrahedral type is studied. Data are presented on primes $v\leq 10^8$ raising the level of $3$-adic even representations of various conductors. Based on the data, a conjecture is formulated concerning the distribution of certain lines in the plane. By an application of Wiles' formula, the conjecture is shown to imply that the density of primes raising the level of a $p$-adic even representation is $$\frac{p-1}{p},$$ in agreement with the density of $2/3$ for $p=3$ observed in the data.

Level-raising of even representations of tetrahedral type and equidistribution of lines in the projective plane

TL;DR

The paper analyzes how often primes can be added to the level of even 3-adic tetrahedral-type Galois representations by linking local ramification data to global Selmer conditions via a balanced deformation framework. It provides extensive computational data for p=3 up to v ≤ 10^8 and proves a general density theorem: if the ramified lines in the local cohomology space are equidistributed in the projective plane, level-raising primes occur with density (p−1)/p, matching observed 2/3 for p=3. A central conjecture with Ramakrishna posits uniform distribution of these lines, connecting local probabilistic behavior to global deformation-theoretic phenomena. The work also discusses the rarity of even representations and the role of Selmer conditions at nice primes, highlighting the interplay between arithmetic geometry, Galois cohomology, and prime distribution in this setting.

Abstract

The distribution of primes raising the level of even Galois representations of tetrahedral type is studied. Data are presented on primes raising the level of -adic even representations of various conductors. Based on the data, a conjecture is formulated concerning the distribution of certain lines in the plane. By an application of Wiles' formula, the conjecture is shown to imply that the density of primes raising the level of a -adic even representation is in agreement with the density of for observed in the data.
Paper Structure (10 sections, 18 theorems, 73 equations, 1 figure, 4 tables)

This paper contains 10 sections, 18 theorems, 73 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Level-raising at primes $v$
  4. Ramification at the auxiliary prime
  5. Descent
  6. Data
  7. Density of level-raising primes for $\operatorname{GL}(2,\mathbb{Z}_p)$
  8. On the rarity of even representations
  9. Selmer conditions at nice primes
  10. Density $(p-1)/p$ and equidistribution of lines in the projective plane

Key Result

Lemma 2

The local cohomology groups at $v$ have the following ranks:

Figures (1)

  • Figure 1: Tetrahedron with symmetry group $A_4$ (Langlands langlands)

Theorems & Definitions (41)

  • Definition 1: Auxiliary primes
  • Lemma 2
  • proof
  • Lemma 3
  • Definition 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 31 more