  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.
