Uniform boundedness of p-primary torsion of abelian schemes

• 成果类型: Article
• 署名作者: Cadoret, Anna; Tamagawa, Akio
• 署名单位: Kyoto University; Centre National de la Recherche Scientifique (CNRS); Inria; Universite de Bordeaux
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-011-0343-6
• 发表日期: 2012
• 页码: 83-125
• 关键词: elliptic-curves; points; VARIETIES; FIELDS

摘要

Let k be a field finitely generated over Q and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d = 1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d = 1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E -> S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E-s, s is an element of S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves. More precisely, we prove the following. Denote by Gamma(k) the absolute Galois roup of k. For an abelian variety A over k and a character chi : Gamma(k) -> Z(p)*, define A[p(infinity)](chi) to be the module of p-primary torsion of A((k) over bar) on which Gamma(k) acts as chi-multiplication. Assume that. does not appear as a subrepresentation of the p-adic representation associated with an abelian variety over k. Then A[p(infinity)](chi) is always finite, but the exponent of A[p(infinity)](chi) may depend on A, a priori. Our main result is about the uniform boundedness of A[p(infinity)](chi) when A varies in a 1-dimensional family. More precisely, if S is a curve over k and A is an abelian scheme over S, then there exists an integer N := N(A, S, k, p, chi), such that A(s) [p(infinity)](chi) subset of A(s) [p(N)] holds for any s is an element of S(k). This arithmetic result is obtained as a corollary of the following geometric result on the p-primary torsion of abelian varieties over function fields of curves, combined with Mordell's conjecture (Faltings' theorem). Let K be the function field of a curve over an algebraically closed field of characteristic 0 and A an abelian variety over K. Assume for simplicity that A contains no nontrivial isotrivial subvariety. Then, for any c >= 0, there exists an integer N := N(c, A, S, k, p) >= 0 such that A[p(infinity)](K') subset of A[p(N)] for all finite extension K'/K with K' of genus <= c. A key ingredient of the proof of this geometric result is a certain result on the number of points on reduction modulo p(n) of p-adic analytic homogeneous spaces. Our uniform boundedness result when chi is the trivial character gives the uniform boundedness for the k-rational p-primary torsion in the fibres A(s), s is an element of S(k) alluded to above. When chi is the p-adic cyclotomic character, together with certain descent methods, it also yields a proof of the 1-dimensional case of (a generalized variant of) the modular tower conjecture, which was, actually, the original motivation for this work. This is a conjecture arising from the regular inverse Galois problem, whose original form was posed by M. Fried in the early 1990s.