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On Hecke eigenvalues of Ikeda lifts

Sanoli Gun, Sunil L Naik

Abstract

A well known result of Breulmann states that Hecke eigenvalues of Saito-Kurokawa lifts are positive. In this article, we show that the Hecke eigenvalues of an Ikeda lift at primes are positive. Further, we derive lower and upper bounds of these Hecke eigenvalues for all primes $p$. One of the main ingredients involves expressing the Hecke eigenvalues of an Ikeda lift in terms of certain reciprocal polynomials.

On Hecke eigenvalues of Ikeda lifts

Abstract

A well known result of Breulmann states that Hecke eigenvalues of Saito-Kurokawa lifts are positive. In this article, we show that the Hecke eigenvalues of an Ikeda lift at primes are positive. Further, we derive lower and upper bounds of these Hecke eigenvalues for all primes . One of the main ingredients involves expressing the Hecke eigenvalues of an Ikeda lift in terms of certain reciprocal polynomials.
Paper Structure (8 sections, 3 theorems, 47 equations)

This paper contains 8 sections, 3 theorems, 47 equations.

Table of Contents

  1. Introduction and Statements of Results
  2. Prerequisites
  3. Prerequisites from $q$-binomial coefficients
  4. Prerequisites from Siegel modular forms
  5. Saito-Kurokawa lift
  6. Ikeda lift
  7. Hecke eigenvalues of Ikeda lifts
  8. Proof of Theorem \ref{['thm4']}

Key Result

Theorem 1

Let $F \in S_k(\Gamma_n)$ be an Ikeda lift with Hecke eigenvalues $\{\lambda_F(m) : m \in \mathbb{N}\}$. Then we have for all primes $p$. Further, we have for all primes $p$.

Theorems & Definitions (6)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Remark 2.1
  • Lemma 3
  • proof