[Paper Review] On the numerator of the symplectic Hecke series of degree three

Alexei Panchishkin, Kirill Vankov|ArXiv.org|Apr 27, 2006

Advanced Algebra and Geometry4 citations

TL;DR

This paper presents a novel method to compute the numerator of the symplectic Hecke series for $\mathrm{Sp}_3$, using Satake spherical maps and symmetric polynomials to derive explicit formulas for the generating series. The key contribution is a degree-6 polynomial $P(x_0,x_1,x_2,x_3,v)$ encoding the Hecke operator images, with coefficients expressed in terms of symmetric polynomials and powers of $p$, confirming Shimura's conjecture via a new computational approach.

ABSTRACT

We find a different method to compute the generating series in Shimura's conjecture for Sp3, proved by Andrianov in 1967. Formulas for the Satake spherical map for Sp3 are used.

Motivation & Objective

• To provide an alternative, systematic method for computing the numerator of the symplectic Hecke series for $\mathrm{Sp}_3$, as conjectured by Shimura.
• To derive explicit formulas for the generating series of Hecke operators on $\mathrm{Sp}_3$ using the Satake spherical map.
• To express the images of Hecke operators under the spherical map in terms of symmetric polynomials and $p$-adic coefficients.
• To verify Shimura's conjecture for $\mathrm{Sp}_3$ through direct computation of the numerator polynomial using symmetric function theory.

Proposed method

• The authors use the Satake spherical map $\Omega$ to translate Hecke operators on $\mathrm{Sp}_3$ into symmetric polynomials in variables $x_1,x_2,x_3$.
• They define symmetric polynomials $\mathrm{sym}_{i_1,i_2,i_3}$ via orbit sums under the symmetric group $S_3$, normalized to have coefficient 1.
• The generating series numerator is constructed as a degree-6 polynomial $P(x_0,x_1,x_2,x_3,v)$ with coefficients involving $\mathrm{sym}_{i_1,i_2,i_3}$ and powers of $p$.
• The method computes $\omega(t(1,p^\lambda_2,p^{\lambda_3}))$ for various $\lambda_2,\lambda_3$ using the spherical map and symmetric matrix counts over $\mathbb{F}_p$, with results expressed as rational functions in $p$.
• Explicit formulas for $\Omega(\mathbf{T}(p))$, $\Omega(\mathbf{T}_1(p^2))$, and $\Omega(\mathbf{T}_2(p^2))$ are derived using $\mathrm{sm}_p(r,a)$, the number of symmetric matrices of rank $r$ over $\mathbb{F}_p$.
• The computation is validated by substituting specific Satake parameters $x_0=1, x_1=p, x_2=p^2, x_3=p^3$, yielding a factorized form of $P_\nu(v)$ that matches known $L$-function factors.

Experimental results

Research questions

• RQ1 How can the numerator of the symplectic Hecke series for $\mathrm{Sp}_3$ be computed using the Satake spherical map and symmetric polynomials?
• RQ2 What is the explicit form of the generating series for Hecke operators on $\mathrm{Sp}_3$ in terms of symmetric polynomials and $p$-adic coefficients?
• RQ3 How do the images of $\mathbf{T}(p)$, $\mathbf{T}_1(p^2)$, and $\mathbf{T}_2(p^2)$ under the spherical map relate to symmetric polynomials and $p$-adic counts?
• RQ4 Can the conjecture of Shimura for $\mathrm{Sp}_3$ be verified through direct computation of the numerator polynomial?
• RQ5 What is the structure of $\omega(t(1,p^{\lambda_2},p^{\lambda_3}))$ for various $\lambda_2, \lambda_3$, and how does it decompose into symmetric polynomial terms?

Key findings

• The numerator of the Hecke-Shimura generating series for $\mathrm{Sp}_3$ is given by a degree-6 polynomial $P(x_0,x_1,x_2,x_3,v)$ with coefficients involving symmetric polynomials $\mathrm{sym}_{i_1,i_2,i_3}$ and powers of $p$, as shown in equation (1).
• The image of $\mathbf{T}(p)$ under the spherical map is $x_0(\mathrm{sym}_{1,1,0} + \mathrm{sym}_{1,0,0} + \mathrm{sym}_{1,1,1} + 1)$, confirming the standard Satake parameterization.
• The image of $\mathbf{T}_1(p^2)$ is expressed as a sum of terms involving $x_0^2$ and symmetric polynomials with $p$-dependent coefficients, such as $\frac{(p^2-1)\mathrm{sym}_{2,1,1}}{p^3}$.
• The image of $\mathbf{T}_2(p^2)$ is computed as $x_0^2\omega(t(1,p,p)) + p^4x_0^2\omega(t(p,p,p^2)) + \mathrm{sm}_p(1,3)x_0^2\omega(t(p,p,p))$, with $\mathrm{sm}_p(1,3)$ evaluated as $\frac{\phi_3(p)}{\phi_1(p)\phi_2(p)}$.
• For the special case $x_0=1, x_1=p, x_2=p^2, x_3=p^3$, the polynomial $P_\nu(v)$ factors as $(1-p v)(1-p^2 v)(1-p^3 v)(1-p^4 v)(1+p v + p^2 v + p^3 v + p^4 v + p^5 v^2)$, confirming consistency with known $L$-function factors.
• The paper provides explicit rational expressions for $\omega(t(1,p^{\lambda_2},p^{\lambda_3}))$ for $\lambda_2,\lambda_3$ from 0 to 6, each decomposed into symmetric polynomial terms with $p$-dependent denominators, such as $\frac{(p^3-p^2)\mathrm{sym}_{6,5,0}}{p^{18}}$ for $\omega(t(1,p^5,p^6))$.