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On divisibility of Hecke eigenvalues of Ikeda lifts

Sanoli Gun, Sunil Naik

TL;DR

This work analyzes when primes divide the p-th Hecke eigenvalues of Ikeda lifts. It connects divisibility questions to the arithmetic of ℓ-adic Galois representations attached to elliptic Hecke eigenforms and uses Chebotarev density to study primes in arithmetic progressions of Fourier coefficients. The authors establish the existence and precise asymptotics of densities δ_F(ℓ^m) for Ikeda lifts, including explicit leading terms and nontrivial bounds, and they identify abelian subfields A_{ℓ^m} lying inside kernel-fixed fields, clarifying the abelian structure of the associated Galois extensions. They also prove distribution results for Fourier coefficients a_f(p) in arithmetic progressions, providing unconditional and GRH-conditioned error terms, and they deduce consequences for the divisibility of Ikeda eigenvalues in terms of these densities. Collectively, the results give a detailed density and distribution picture for primes governed by Ikeda lifts, informing both theoretical understanding and potential applications in automorphic form arithmetic.

Abstract

In this article, we estimate the density of the set of primes $p$ such that the $p$-th Hecke eigenvalue of an Ikeda lift is divisible by a fixed positive integer. One of the main ingredients involves the study of abelian subfields of fixed fields of the kernel of Galois representations attached to elliptic Hecke eigenforms. Further, we study the distribution of Fourier coefficients of elliptic Hecke eigenforms in arithmetic progressions.

On divisibility of Hecke eigenvalues of Ikeda lifts

TL;DR

This work analyzes when primes divide the p-th Hecke eigenvalues of Ikeda lifts. It connects divisibility questions to the arithmetic of ℓ-adic Galois representations attached to elliptic Hecke eigenforms and uses Chebotarev density to study primes in arithmetic progressions of Fourier coefficients. The authors establish the existence and precise asymptotics of densities δ_F(ℓ^m) for Ikeda lifts, including explicit leading terms and nontrivial bounds, and they identify abelian subfields A_{ℓ^m} lying inside kernel-fixed fields, clarifying the abelian structure of the associated Galois extensions. They also prove distribution results for Fourier coefficients a_f(p) in arithmetic progressions, providing unconditional and GRH-conditioned error terms, and they deduce consequences for the divisibility of Ikeda eigenvalues in terms of these densities. Collectively, the results give a detailed density and distribution picture for primes governed by Ikeda lifts, informing both theoretical understanding and potential applications in automorphic form arithmetic.

Abstract

In this article, we estimate the density of the set of primes such that the -th Hecke eigenvalue of an Ikeda lift is divisible by a fixed positive integer. One of the main ingredients involves the study of abelian subfields of fixed fields of the kernel of Galois representations attached to elliptic Hecke eigenforms. Further, we study the distribution of Fourier coefficients of elliptic Hecke eigenforms in arithmetic progressions.

Paper Structure

This paper contains 13 sections, 5 theorems, 116 equations, 2 figures.

Table of Contents

  1. Introduction and Statements of Results
  2. Prerequisites
  3. Prerequisites from Siegel modular forms
  4. Saito-Kurokawa lifts
  5. Ikeda lifts
  6. Applications of $\ell$-adic Galois representations and Chebotarev density theorem
  7. The image of Galois representation $\rho_{\ell, f}$
  8. Intersection of the fields $K_{\ell^m}$ and $\mathbb{Q}$ζ_ℓ^m$$
  9. Proof of Theorem \ref{['thm4']}
  10. Proof of Lemma \ref{['LAlm']}
  11. Distribution of Fourier coefficients in Arithmetic progressions
  12. Proof of Theorem \ref{['thm3']}
  13. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1

Let $F \in S_k(\Gamma_n)$ be an Ikeda lift of a normalized Hecke eigenform in $S_{2k-n}(\Gamma_1)$ with integer Fourier coefficients. Also let $\ell$ be a prime number and $m$ be a positive integer. Then the density of the set $\Pi_F(\ell^m)$ exists, denoted by $\delta_F(\ell^m)$, and we have Further, there exists a positive constant $c$ such that uniformly for $\ell^m \leq (\log x)^{c}$. Moreov

Figures (2)

  • Figure 1: Intersection of $K_{\ell^m}$ and $\mathbb{Q}$ζ_ℓ^m$$
  • Figure 2: Compositum of $K_{\ell^m}$ and $\mathbb{Q}$ζ_ℓ^m$$

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Theorem 4
  • Lemma 5
  • proof