Courbes algébriques et équations multiplicatives
Courbes algébriques et équations multiplicatives
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代数和乘法方程
DOI：
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发表时间：
2008
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通讯作者：
G. Maurin
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文献类型：
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作者：
G. Maurin
我们研究位于 $${ ar{mathbb{Q}}}$$ 上的乘法圆环中的代数曲线 C 与余维 2 的所有代数子群的并集的交集。Bombieri、Masser 和 Zannier 已经在假设 C 不包含在真子圆环的平移中的假设下证明了该集合的有限性。根据这个结果，这些作者提出了隐含有限性的最小假设的问题，并引发了猜想：当C不包含在真子群中时，有限性恰好成立。我们在这里证明这个陈述，它也是 Zilber 和 Pink 独立陈述的更一般猜想的一个特例。我们的证明的灵感来自于 Rémond 和 Viada 的一篇文章，该文章涉及椭圆曲线幂中的曲线的 Zilber-Pink 猜想。因此，它依赖于通过雷蒙德广义 Vojta 不等式证明的 Vojta 不等式的统一版本。主要任务是为某些交叉数建立下界，这里是通过炸毁 C × C 的紧致化而获得的整个曲面族。
We study the intersection of an algebraic curve C lying in a multiplicative torus over $${ar{mathbb{Q}}}$$ with the union of all algebraic subgroups of codimension 2. Finiteness of this set has already been proved by Bombieri, Masser and Zannier under the assumption that C is not contained in a translate of a proper subtorus. Following this result, the question of the minimal hypothesis implying finiteness has been raised by these authors, giving rise to the conjecture~: finiteness holds precisely when C is not contained in a proper subgroup. We prove here this statement which is also a special case of more general conjectures stated independently by Zilber and Pink. Our proof takes its inspiration from an article by Rémond and Viada concerning the Zilber-Pink conjecture for curves lying in a power of an elliptic curve. Hence, it relies on a uniform version of the Vojta inequality proven via the generalized Vojta inequality of Rémond. The main task is to establish a lower bound for some intersection numbers, here on a whole family of surfaces obtained by blowing up a compactification of C × C.