2004/05/23 by Dmitry Logachev, Logachev, Dmitry · 1 citation
Mathematics · #11G18 #14G35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT #msc:11G18 #msc:14G35
paper · pdf · doi:10.48550/arxiv.math/0405442
26 pages
arxiv created 2004/05/23 · openalex publication_date 2004/05/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.