2023/06/04 by Xiaoyu Zhang, Zhang, Xiaoyu
Computer Science · Mathematics · #11G15 #14K22 #22D40 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2306.02290
openalex publication_date 2023/06/04 · openalex created_date 2023/06/07 · openalex updated_date 2026/07/28
For a point x0 in a Shimura variety attached to a Shimura datum of Hodge type (G,X), we have an associated abelian scheme A0. Fixing a non-empty finite set S of primes, we consider the simultaneous supersingular reduction modulo ℓ\inS of (several copies of) p-adic Hecke orbits of A0. We give a precise description of the image of this map. As an application, we give a more conceptual proof of Mazur's conjecture on non-torsionness of higher Heegner points on an abelian variety which is a quotient of the Jacobian of a Shimura curve. Our arguments simplify those of C.Cornut and V.Vatsal in two important aspects: (1) we do not need to assume the p-adic group G1(ℚp)/ZG1(ℚp) to be simple; (2) we do not need to consider separately the ``geometric" part and ``chaotic" part in the Hecke orbits.