Relations between conjectural eigenvalues of Hecke operators on submotives of Siegel varieties

TL;DR

This paper explores conjectural relations between eigenvalues of Hecke operators on submotives of Siegel varieties, revealing linear relations with polynomial coefficients, as a step toward generalizing Kolyvagin's theorem to higher-dimensional Shimura varieties.

Contribution

It derives explicit, conjectural linear relations between Hecke eigenvalues on submotives of Siegel varieties, with polynomial coefficients satisfying recurrence formulas.

Findings

Relations are linear with polynomial coefficients in p.

Coefficients satisfy a simple recurrence formula.

Results are applicable to general Shimura varieties.

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.