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

D. Yu. Logachev

Dalʹnevostočnyj matematičeskij žurnal, Tome 12 (2012) no. 1, pp. 60-85

Résumé

There exist conjectural formulas of 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 where $X$ is a Siegel variety. Hence, this result is conjectural as well: the 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 and 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 the generalization of Kolyvagin's theorem of the finiteness of Tate – Shafarevich group of elliptic curves of analytic rank 0 and 1 over $\mathbb Q$, to the case of submotives of other Shimura varieties, particularly of Siegel varieties of genus 3, see [9]. The idea of the proof: on the one hand, the above formulas of Langlands – Arthur give us (conjectural) relations between Weil numbers of a submotive. On the other hand, the Satake map permits us to transform these relations between Weil numbers into relations between eigenvalues of $p$-Hecke operators on $X$. The paper also contains a survey of some related questions, for example explicit finding of the Hecke polynomial for $X$, and (Appendix) tables for the cases $g=2,3$.