Hecke Eigenvalues of Ikeda Lifts

Technical Summary: Hecke Eigenvalues of Ikeda Lifts

Problem Statement
The paper investigates the Hecke eigenvalues $\lambda_F(T)$ associated with Ikeda lifts, which are Siegel cusp forms of genus $2n$ constructed from elliptic modular forms of genus 1. While it is established that the Hecke eigenvalues $\lambda_F(p)$ for the operator $T(p)$ are positive for Ikeda lifts, the positivity of eigenvalues $\lambda_F(p^r)$ for higher powers $r \ge 1$ remains a non-trivial question. Previous approaches relied on Dirichlet series relations involving the spin $L$-function, specifically analyzing the numerator polynomial $P(x)$ in the local factor expansion. However, explicit formulas for this numerator are only known for low genera ($n \le 4$), limiting the ability to generalize positivity results to arbitrary genus $n$. The authors aim to establish the positivity of $\lambda_F(p^r)$ for all $n$ without relying on the explicit form of the Dirichlet series numerator.

Methodology
The authors adopt a direct algebraic approach by computing the spherical image of the Hecke operators $T(p^r)$ under the spherical map $\Omega$. The methodology proceeds as follows:

1. Spherical Map Construction: The paper utilizes the spherical map $\Omega: H_{n,p} \to \mathbb{Q}[x_0, \dots, x_n]^{W_n}$, which maps local Hecke algebra elements to Weyl group-invariant polynomials. The eigenvalue $\lambda_F(T)$ is obtained by evaluating $\Omega(T)$ at the Satake $p$-parameters of the form $F$.
2. Reduction to General Linear Group: The authors relate the spherical map for the symplectic group to that of the general linear group. They utilize an explicit formula for the spherical image of $T(p^r)$ in the general linear case (derived from convolution algebra representations and symmetric functions) to construct the symplectic formula.
3. Specialization to Ikeda Lifts: The core technical innovation lies in substituting the specific Satake parameters of an Ikeda lift into the general symplectic formula. The Satake parameters of the Ikeda lift $F$ are derived from those of the underlying elliptic form $f$ (denoted $a, a^{-1}$).
4. Vanishing Terms: A critical step involves analyzing the sum over the symmetric group $S_{2n}$ present in the general formula. The authors prove that for Ikeda lifts, the coefficients $c(\sigma(a_1, \dots, a_{2n}))$ vanish for all permutations $\sigma \neq \text{id}$. This drastic simplification reduces the complex sum to a single term corresponding to the identity permutation.
5. Polynomial Analysis: The resulting explicit formula for $\lambda_F(p^r)$ is expressed as a polynomial in $p^{\pm 1/2}$. The authors analyze the coefficients of this polynomial, establishing bounds on their magnitude and identifying the leading term.

Key Contributions and Results

• Explicit Formula: The paper derives a closed-form expression for the Hecke eigenvalues $\lambda_F(p^r)$ of an Ikeda lift $F \in S_{k+n}(\Gamma_{2n})$ in terms of the Satake parameters $a, a^{-1}$ of the generating elliptic form $f$.
  $$\lambda_F(p^r) = p^{r(nk - \frac{n}{2})} \sum_{0 \le \delta_1 \le \dots \le \delta_{2n} \le r} a^{-nr + \sum \delta_i} p^{\sum (-n-\frac{1}{2}+i)\delta_i} \frac{\phi_{2n}(p^{-1})}{\phi_{k_1}(p^{-1})\dots\phi_{k_t}(p^{-1})}$$
  where $\phi_m$ are specific polynomials related to the structure of the Hecke algebra.

• Positivity for Large Primes: By analyzing the polynomial structure, the authors prove that $\lambda_F(p^r)$ can be written as a polynomial in $p^{1/2}$ with integer coefficients in $\mathbb{Z}[a, a^{-1}]$. They demonstrate that the leading coefficient is 1 and that the remaining coefficients are bounded. Consequently, for any fixed $r \ge 1$, the eigenvalue $\lambda_F(p^r)$ is strictly positive for all sufficiently large primes $p$.

• Explicit Lower Bound: The paper provides a concrete lower bound for the primes $p$ ensuring positivity. Specifically, $\lambda_F(p^r) > 0$ for all:
  $$p > \left( 4n \left( 2rn^2 + 2n(2n-1) \right) \right)^2$$
  This bound is derived from the precise estimates of the polynomial coefficients obtained in Proposition 3.

Significance
The paper claims significance by providing a method to determine the positivity of Hecke eigenvalues for Ikeda lifts of arbitrary genus without relying on the explicit knowledge of the numerator polynomial in the spin $L$-function Dirichlet series. This circumvents the limitation of previous results which were restricted to genus $n \le 4$. The result generalizes Breulmann's theorem (which linked positivity of eigenvalues to the Saito-Kurokawa lift for genus 2) to the broader class of Ikeda lifts for any genus $2n$. The work establishes that, similar to the genus 2 case, the Hecke eigenvalues of Ikeda lifts are positive for large primes, offering further evidence regarding the arithmetic properties of these lifts.