A combinatorial Li-Tau inequality and rational points on curves
A combinatorial Li-Tau inequality and rational points on curves
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组合 Li-Tau 不等式和曲线上的有理点
DOI：
10.1007/s00208-014-1067-x
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Janne Kool
中科院分区：
文献类型：
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作者：
Gunther Cornelissen;Fumiharu Kato;Janne Kool
提出了一种基于图论的非阿基米德曲线的角度控制方法。记为完备非阿基米德值域。本文首先证明了曲线在代数闭包上的一致性的一个下界，即：曲线稳定模型的对偶图的任何加细所产生的到树的有限调和图态射，都应该最小化。接下来是我们的主要结果：我们证明了一个下界的程度，这样的一个图形态射的拉普拉斯算子的第一个特征值和一些“体积”的原始图形，这可以被看作是一个替代图的李丘不等式从微分几何，虽然我们也证明了严格的模拟原始不等式失败的一般图形。最后，我们应用的结果，给出了一个下界的任意Drinfeld模曲线在有限域和一般的同余子群softhat是线性的指数，一个常数，只取决于剩余领域的程度和程度的选择“无限”的地方。这是一个功能领域的模拟定理阿布拉莫维奇经典的模块化曲线。我们目前的应用程序，以统一的有界性的扭转秩两Drinfeld模块，改善现有的结果，并在一定的椭圆曲线的模块化程度上的函数域，解决问题的Papikian下界。
We present a method to control gonality of nonarchimedean curves based on graph theory. Letdenote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a curve over the algebraic closure ofin terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from anyrefinementof the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen as a substitute for graphs of the Li–Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally, we apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroupsofthat is linear in the index, with a constant that only depends on the residue field degree and the degree of the chosen “infinite” place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian.