Mathematics – Algebraic Geometry
Scientific paper
2004-05-23
Mathematics
Algebraic Geometry
26 pages
Scientific paper
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.
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