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Comparing Hecke eigenvalues for pairs of automorphic representations for GL(2)

Kin Ming Tsang

TL;DR

The article investigates how local comparisons of Hecke eigenvalues for pairs of GL2 automorphic representations constrain global isomorphism, focusing on the lower Dirichlet density of places where $|a_v(\pi_1)|$ and $|a_v(\pi_2)|$ differ or satisfy inequalities. By analyzing products and adjoint lifts via Rankin–Selberg and symmetric power L-functions, and classifying pairs into dihedral and non-dihedral (including solvable polyhedral types), the authors prove a robust lower bound $\underline{\delta}(S_*^{>}(\pi_1,\pi_2)) \geq \tfrac{1}{16}$ for non-twist-equivalent pairs, and derive refined bounds in dihedral and mixed-dihedral cases. The results also yield improvements over prior bounds (e.g., Wong) for densities of places where $|a_v(\pi_1)| \neq |a_v(\pi_2)|$, and the paper demonstrates sharpness of several bounds through explicit tetrahedral and dihedral examples. Overall, the work advances quantitative strong multiplicity-type statements for GL2 by leveraging automorphy of symmetric powers and adjoint lifts, with potential implications for understanding global equivalence from local data. $L$-functions and density estimates are presented with precise pole-structure analyses at $s=1$, tying local trace inequalities to global arithmetic in a nuanced, case-by-case framework.

Abstract

We consider a variant of the strong multiplicity one theorem. Let $π_{1}$ and $π_{2}$ be two unitary cuspidal automorphic representations for $\mathrm{GL(2)}$ that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(π_{1}) \right\rvert > \left\lvert a_{v}(π_{2}) \right\rvert$, where $a_{v}(π_{i})$ is the trace of Langlands conjugacy class of $π_{i}$ at $v$. One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(π_{1})\right\rvert \neq \left\lvert a_{v}(π_{2}) \right\rvert$.

Comparing Hecke eigenvalues for pairs of automorphic representations for GL(2)

TL;DR

The article investigates how local comparisons of Hecke eigenvalues for pairs of GL2 automorphic representations constrain global isomorphism, focusing on the lower Dirichlet density of places where and differ or satisfy inequalities. By analyzing products and adjoint lifts via Rankin–Selberg and symmetric power L-functions, and classifying pairs into dihedral and non-dihedral (including solvable polyhedral types), the authors prove a robust lower bound for non-twist-equivalent pairs, and derive refined bounds in dihedral and mixed-dihedral cases. The results also yield improvements over prior bounds (e.g., Wong) for densities of places where , and the paper demonstrates sharpness of several bounds through explicit tetrahedral and dihedral examples. Overall, the work advances quantitative strong multiplicity-type statements for GL2 by leveraging automorphy of symmetric powers and adjoint lifts, with potential implications for understanding global equivalence from local data. -functions and density estimates are presented with precise pole-structure analyses at , tying local trace inequalities to global arithmetic in a nuanced, case-by-case framework.

Abstract

We consider a variant of the strong multiplicity one theorem. Let and be two unitary cuspidal automorphic representations for that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which , where is the trace of Langlands conjugacy class of at . One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which .
Paper Structure (15 sections, 13 theorems, 69 equations)

This paper contains 15 sections, 13 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Automorphic L-functions
  4. Rankin-Selberg L-functions
  5. Symmetric powers
  6. Isobaric representations and adjoint lift
  7. Solvable polyhedral representation
  8. Cuspidality of symmetric powers
  9. Both $\pi_{1}$ and $\pi_{2}$ are non-dihedral
  10. Both $\pi_{1}$ and $\pi_{2}$ are dihedral
  11. Exactly one of $\pi_{1}$ and $\pi_{2}$ is dihedral
  12. Examples
  13. Tetrahedral example
  14. Dihedral example 1
  15. Dihedral example 2

Key Result

Theorem 1.1

Let $\pi_{1}$ and $\pi_{2}$ be cuspidal automorphic representations for $\mathop{\mathrm{GL}}\nolimits_{2}(\mathbb{A}_{F})$ with unitary central characters. Assume that $\pi_{1}$ and $\pi_{2}$ are not twist-equivalent. Then If we further assume that both $\pi_{1}$ and $\pi_{2}$ are non-solvable polyhedral, then

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Remark
  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 17 more