[Paper Review] Relations between conjectural eigenvalues of Hecke operators on submotives of Siegel varieties

Dmitry Logachev|ArXiv.org|May 23, 2004

Advanced Algebra and Geometry5 references3 citations

TL;DR

This paper conjecturally derives linear relations between eigenvalues of $p$-Hecke operators on cohomology of submotives of Siegel varieties, using Langlands–Arthur conjectures and the Satake map. The key result is that these relations are polynomial in $p$ and satisfy a recurrence, with explicit formulas provided for genus $g=2,3$ via Weil number computations and Hecke polynomial factorization.

ABSTRACT

There exist conjectural formulas on relations between $L$-functions of submotives of Shimura varieties and automorphic representations of the corresponding reductive groups, due to Langlands -- Arthur. In the present paper these formulas are used in order to get explicit relations between eigenvalues of $p$-Hecke operators (generators of the $p$-Hecke algebra of $X$) on cohomology spaces of some of these submotives, for the case $X$ is a Siegel variety. Hence, this result is conjectural as well: methods related to counting points on reductions of $X$ using the Selberg trace formula are not used. It turns out that the above relations are linear, their coefficients are polynomials in $p$ which satisfy a simple recurrence formula. The same result can be easily obtained for any Shimura variety. This result is an intermediate step for a generalization of the Kolyvagin's theorem of finiteness of Tate -- Shafarevich group of elliptic curves of analytic rank 0, 1 over $Q$, to the case of submotives of other Shimura varieties, particularly of Siegel varieties of genus 3.

Motivation & Objective

• To establish conjectural linear relations between eigenvalues of $p$-Hecke operators on cohomology of submotives of Siegel varieties.
• To generalize Kolyvagin’s finiteness theorem for Tate–Shafarevich groups to submotives of Siegel varieties of genus 3.
• To derive explicit polynomial relations in $p$ for Hecke eigenvalues using Weil numbers and the Satake map.
• To provide computational tools and tables for $g=2$ and $g=3$ as a foundation for further study of automorphic $L$-functions on Shimura varieties.

Proposed method

• Leverages Langlands–Arthur conjectures to relate $L$-functions of submotives to automorphic representations on reductive groups.
• Applies the Satake map to translate relations between Weil numbers into relations between $p$-Hecke operator eigenvalues.
• Uses the structure of parabolic subgroups $P$ in $GSp_{2g}$ to classify and compute eigenvalue relations for submotives.
• Derives recurrence relations for polynomial coefficients of the eigenvalue relations via symmetric polynomials of Weil numbers.
• Performs explicit calculations for $g=2$ and $g=3$, including factorization of Hecke polynomials and substitution of trace relations.
• Provides tables in the appendix for $g=2$ and $g=3$ to support numerical verification and future applications.

Experimental results

Research questions

• RQ1 What are the conjectural linear relations between eigenvalues of $p$-Hecke operators on cohomology of submotives of Siegel varieties?
• RQ2 How do Weil number relations induced by Langlands–Arthur conjectures translate into Hecke eigenvalue relations via the Satake map?
• RQ3 Can a recurrence formula be derived for the polynomial coefficients of these eigenvalue relations in terms of $p$?
• RQ4 What explicit forms do these relations take for Siegel varieties of genus $g=2$ and $g=3$?
• RQ5 How can these relations be used to generalize Kolyvagin’s finiteness theorem to higher genus Siegel varieties?

Key findings

• The relations between $p$-Hecke eigenvalues $\goth m_{p,i}$ are linear and their coefficients are polynomials in $p$ satisfying a simple recurrence formula.
• For $g=3$, the Hecke polynomial is expressed in terms of $\tau_p$ and $\tau_{p,1}$, with explicit coefficients involving $p^2$, $p^5$, and $p^{10}$.
• The relation $A = 0$ holds, where $A$ is a combination of $\goth m_p$, $\goth m_{p,1}$, $\goth m_{p,2}$, and polynomials in $p$, derived from $\goth h_3 = 0$.
• The symmetric sum $\sigma_2(\alpha_*)$ simplifies to $p^2(\goth m_{p,2} - p^4 + p^3 - p^2 + 1)$ under the condition $A=0$, reducing complexity.
• The Hecke polynomial for $g=3$ is obtained by substituting $\tau_{p,2}$ from (4.5) and factoring, yielding a degree-4 polynomial in the Satake parameter.
• Tables in the appendix provide explicit values of $\goth m_{p,i}$ and Weil numbers for $g=2$ and $g=3$, supporting numerical and theoretical extensions.