Uniform boundedness of p-primary torsion of abelian schemes

Abstract

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∈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 Γ k the absolute Galois group of k. For an abelian variety A over k and a character $\chi:\Gamma_{k}\rightarrow\mathbb{Z}_{p}^{\ast}$ , define A[p ∞](χ) to be the module of p-primary torsion of $A(\overline{k})$ on which Γ k acts as χ-multiplication. Assume that χ does not appear as a subrepresentation of the p-adic representation associated with an abelian variety over k. Then A[p ∞](χ) is always finite, but the exponent of A[p ∞](χ) may depend on A, a priori. Our main result is about the uniform boundedness of A[p ∞](χ) 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,χ), such that A s [p ∞](χ)⊂A s [p N ] holds for any s∈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 ∞](K′)⊂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 χ is the trivial character gives the uniform boundedness for the k-rational p-primary torsion in the fibres A s , s∈S(k) alluded to above. When χ 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.