DIOPHANTINE APPROXIMATION ON ABELIAN VARIETIES IN CHARACTERISTIC p
DIOPHANTINE APPROXIMATION ON ABELIAN VARIETIES IN CHARACTERISTIC p
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特征p中阿贝尔变种的丢番图逼近
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发表时间：
1995
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通讯作者：
J. Voloch
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作者：
J. Voloch
设A是有限域K上一元变元函数域K上的交换簇，v是K的位置.本文研究A(Kv)上的v-进拓扑导出的A(K)上的拓扑.在许多情况下，这将导致A(K)中的点之间的v进距离以其高度为界，并导致Leopoldt猜想的阿贝尔类似。本文还研究了具有正特征的Abel变种的仿射开子集上的积分点问题。在数域的经典情形下，Lang猜想和Faltings[F]证明了对A是数域K上的交换簇，如果V是A的仿射开集，S是K的有限位置集，则V的S积分点集是有限的。Faltings也有一个无效的界限。Buium和作者[BV]对特征为零的函数场得到了类似的结果，但在正特征中，除了1维的椭圆曲线外，这个问题还没有被研究过。在这种情况下，我们在[V]中得到了关于这个问题的结果。在这篇文章中，我们将得到任意维阿贝尔簇在限制性但相当一般的假设下的一个结果，并在这些假设下推出仿射子集上的积分点的有限性。我们的战略将类似于[BV]。积分点的问题涉及到有理数点到除数的v-adi距离的估计，而这又将通过到较小维度的子簇的距离来估计。在目前的情况下，这一归纳过程是由Hrushovski([H]，见下面的定理1)的一般结果提供的，并将问题归结为估计点之间的v-adi距离的基本问题。我们不能全面地解决我们的问题。我们的结果仅限于当A具有“足够一般的模”(在某种意义上将在下文中详细描述)或者是定义在有限域上的椭圆曲线的情况。中间案件仍在审理中。
Let A be an abelian variety over a function field K in one variable over a finite field k. Let v be a place of K. In this paper we will study the topology induced on A(K) by the v-adic topology on A(Kv). In many cases this will lead to bounds for the v-adic distance between points in A(K) in terms of their height and to abelian analogues of Leopoldt’s conjecture. This paper also studies the question of integral points on affine open subsets of Abelian varieties in positive characteristics. In the classical case of number fields, Lang has conjectured and Faltings [F] proved that for A be an abelian variety over the number field K, if V is an affine open subset of A and S is a finite set of places of K, then the set of S-integral points of V is finite. Faltings has also a non-effective bound. Buium and the author [BV] obtained a similar result for function fields of characteristic zero, but in positive characteristic the problem has not been studied, except in dimension 1, i.e., for elliptic curves. In this case we have obtained, in [V], results on this problem. In this paper we will obtain a result under restrictive, but quite general, hypotheses for abelian varieties of arbitrary dimension and deduce the finiteness of integral points on affine subsets under these hypotheses. Our strategy will be similar to [BV]. The question of integral points is related to estimates for the v-adic distance from rational points to a divisor and this in turn will be estimated by the distance to a subvariety of smaller dimension. This inductive procedure, in the present situation, is provided by a general result of Hrushovski ([H], see theorem 1 below) and reduces the problem to our basic question of estimating v-adic distance between points. We do not solve our problem in full generality. Our results are restricted to the cases when either A has ”sufficiently general moduli”(in a sense that will be precised below) or is an elliptic curve defined over a finite field. The intermediate cases are still open.