Hecke Eigenvalues of Ikeda Lifts

TL;DR

This paper derives explicit formulas for Hecke eigenvalues of Ikeda lifts using the spherical map, demonstrating their positivity for large primes and establishing bounds on their polynomial coefficients.

Contribution

It provides a new explicit formula for Hecke eigenvalues of Ikeda lifts and proves their positivity for large primes.

Findings

Explicit formula for $\lambda_F(p^r)$ in terms of $p^{rac{1}{2}}$

Coefficients of the polynomial are bounded

Hecke eigenvalues are positive for sufficiently large primes

Abstract

In this paper, we study the Hecke eigenvalues of Ikeda lifts. Using the spherical map for the Hecke algebra of the symplectic group, we obtain an explicit formula for the eigenvalues $\lambda_F(p^r)$. From this formula, we show that $\lambda_F(p^r)$ can be written as a polynomial in $p^{\pm 1/2}$ with a positive leading term. Furthermore, we prove that the coefficients of this polynomial are bounded and, as a consequence, the Hecke eigenvalues $\lambda_F(p^r)$ are positive for all sufficiently large primes $p$.