Uniform boundedness of p-primary torsion of abelian schemes
Uniform boundedness of p-primary torsion of abelian schemes
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DOI：
10.1007/s00222-011-0343-6
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发表时间：
2012-04
影响因子：
3.1
通讯作者：
A. Cadoret;Akio Tamagawa
中科院分区：
文献类型：
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作者：
A. Cadoret;Akio Tamagawa
设 k 是一个在 ℚ 和 pa 素数上有限生成的域。阿贝尔簇的扭转猜想（分别为 p-主扭转猜想）overk 预测 ad 维阿贝尔簇 Aoverk 的 k 有理扭转（分别为 p-主 k-有理扭转）应该仅在 kandd 方面有界。这些猜想只知道ford=1。 Thep-primary case 由 Y. Manin 于 1969 年证明；在 B. Mazur、S. Kamienny 等人的一系列贡献之后，L. Merel 于 1996 年完成了一般情况。由于椭圆曲线的模量是一维的，因此扭转猜想（resp.p-主扭转猜想）的 d=1 情况与以下密切相关。对于任何 k 曲线和椭圆方案 E → S，k 有理挠率（分别为 p 主 k 有理挠率）在纤维 E s 中均匀有界，s ∈ S（k）。在本文中，我们将 p 初等情况下的结果扩展到曲线上的任意阿贝尔格式。更准确地说，我们证明了以下内容。用 Гk 表示 k 的绝对伽罗瓦群。对于阿贝尔簇 Aoverk 和一个字符，定义 A[p∞](χ) 为 p 初阶挠率的模，其中 Γk 充当 χ 乘法。假设 χ 不作为与阿贝尔变体 overk 相关的主题表示的子表示出现。那么 A[p∞](χ) 总是有限的，但是 A[p∞](χ) 的指数可能先验地取决于 A。我们的主要结果是关于在一维族中 A 变化时 A[p∞](χ) 的一致有界性。更准确地说，如果S是k上的曲线且A是S上的阿贝尔格式，则存在整数N:=N(A,S,k,p,χ)，使得As[p∞](χ)⊂As[pN]对于anys∈S(k)成立。该算术结果是作为以下关于函数域上阿贝尔簇的p初阶扭转的几何结果的推论而获得的 曲线，结合莫德尔猜想（法尔廷斯定理）。令 K 为特征为 0 的代数闭域上的曲线的函数域，并且 A 为 K 上的阿贝尔簇。为简单起见，假设 A 不包含非平凡的 isotrivial 子品种。那么，对于任何c≥0，存在一个整数N:=N(c,A,S,k,p)≥0，使得A[p∞](K′)⊂A[pN]对于所有有限扩张K′/K和K′≤c。证明这个几何结果的一个关键因素是关于约简模 p-adic 解析齐次空间上的点数的确定结果。当 χ 是平凡特征时，我们的一致有界性结果给出了纤维中 k-有理 ap-主扭转的一致有界性，如上面提到的 s ∈ S(k) 。当χ是p-adic分圆特征时，与某些下降方法一起，它还产生了模塔猜想（的广义变体）的一维情况的证明，这实际上是这项工作的原始动机。这是由正则伽罗瓦逆问题产生的猜想，其原始形式是由 M. Fried 在 20 世纪 90 年代初提出的。
Letkbe a field finitely generated over ℚ andpa prime. The torsion conjecture (resp.p-primary torsion conjecture) for abelian varieties overkpredicts that thek-rational torsion (resp. thep-primaryk-rational torsion) of ad-dimensional abelian varietyAoverkshould be bounded only in terms ofkandd. These conjectures are only known ford=1. Thep-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, thed=1 case of the torsion conjecture (resp.p-primary torsion conjecture) is closely related to the following. For anyk-curveSand elliptic schemeE→S, thek-rational torsion (resp. thep-primaryk-rational torsion) is uniformly bounded in the fibresEs,s∈S(k). In this paper, we extend this result in thep-primary case to arbitrary abelian schemes over curves.More precisely, we prove the following. Denote by Γkthe absolute Galois group ofk. For an abelian varietyAoverkand a character, defineA[p∞](χ) to be the module ofp-primary torsion ofon which Γkacts asχ-multiplication. Assume thatχdoes not appear as a subrepresentation of thep-adic representation associated with an abelian variety overk. ThenA[p∞](χ) is always finite, but the exponent ofA[p∞](χ) may depend onA, a priori. Our main result is about the uniform boundedness ofA[p∞](χ) whenAvaries in a 1-dimensional family. More precisely, ifSis a curve overkandAis an abelian scheme overS, then there exists an integerN:=N(A,S,k,p,χ), such thatAs[p∞](χ)⊂As[pN] holds for anys∈S(k).This arithmetic result is obtained as a corollary of the following geometric result on thep-primary torsion of abelian varieties over function fields of curves, combined with Mordell’s conjecture (Faltings’ theorem). LetKbe the function field of a curve over an algebraically closed field of characteristic 0 andAan abelian variety overK. Assume for simplicity thatAcontains no nontrivial isotrivial subvariety. Then, for anyc≥0, there exists an integerN:=N(c,A,S,k,p)≥0 such thatA[p∞](K′)⊂A[pN] for all finite extensionK′/KwithK′ of genus ≤c. A key ingredient of the proof of this geometric result is a certain result on the number of points on reduction modulopnofp-adic analytic homogeneous spaces.Our uniform boundedness result whenχis the trivial character gives the uniform boundedness for thek-rationalp-primary torsion in the fibresAs,s∈S(k) alluded to above. Whenχis thep-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.