On Signs of Hecke eigenvalues of Ikeda lifts

TL;DR

The paper investigates the sign behavior of Hecke eigenvalues for Ikeda lifts of Siegel modular forms. By relating Hecke eigenvalues to spin L-functions via Schmidt's formulas and Andrianov's generating functions, it proves that for any genus, the eigenvalue at primes $\lambda_{F_f}(p)$ is nonnegative for all sufficiently large primes $p$, and in genus four, the eigenvalues at prime powers $\lambda_{F_f}(p^r)$ are nonnegative for large $p$ with a fixed $r$. The genus-4 case is made explicit through Vankov's polynomials in the Hecke algebra and a partial fraction analysis of $1/Q_{F_f,p}(x)$, yielding concrete positivity results for all primes and asymptotic positivity for $p^r$. These results enhance understanding of the sign patterns of eigenvalues for Ikeda lifts and contribute to the broader study of automorphic forms and their L-functions.

Abstract

Let $F$ be an Ikeda lift, $λ_F(m)$ be the eigenvalue corresponding to the Hecke operator $T(m)$. We show that $λ_F(p)$ is positive for all large enough primes $p$. This is proved for Ikeda lifts of all genus. The second result is specific for genus 4 Ikeda lifts. If $F$ is genus 4 Ikeda lift, we show that, given $r$ there exists a constant $c_r$ such that $λ_F(p^r)$ is positive for all $p>c_r$.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem : Ikeda
• Theorem 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3