论文标题
准标记和形式分析算术表面
Quasi-projective and formal-analytic arithmetic surfaces
论文作者
论文摘要
本回忆录致力于对形式分析算术表面的研究。在Arakelov几何形状的背景下，这些是沿射影复杂曲线的光滑复合分析表面的细菌的算术对应物。正式的分析表面为算术代数定理提供了自然框架，该框架是新旧的。 形式分析算术表面接收了丰富的几何形状，与复杂分析表面的几何形状相同。值得注意的是，伪通道和假腔之间的二分法在其几何形状中起着核心作用。 我们对形式分析算术表面的研究至关重要地依赖于实用值不变的使用。其中一些是相交理论的，本着阿拉克洛夫交集理论的精神。其他一些不变的数字涉及数字的无限几何形状。 将我们的新交叉路口理论与更古典的不变性相关联，这使我们研究了一个新的不变式Archimedean Overflow，该新不变性层连接到从尖锐的紧凑型Riemann表面的分析图上，并与Riemann表面边界。它与Nevanlinna理论的特征功能有关。 我们关于形式分析算术表面的几何形状的结果，允许在算术几何形状的具体问题上应用。值得注意的是，我们概括了Calegari-Dimitrov-Tang的算术全体性定理，该定理涉及功率序列的尺寸，其积分系数满足某些收敛条件。 We also establish an arithmetic counterpart of theorems of Lefschetz and Nori by providing a bound on the index, in the étale fundamental group of an arithmetic surface, of the closed subgroup generated by the étale fundamental groups of some arithmetic curve and of some compact Riemann surfaces mapping to the arithmetic surface.
This memoir is devoted to the study of formal-analytic arithmetic surfaces. These are arithmetic counterparts, in the context of Arakelov geometry, of germs of smooth complex-analytic surfaces along a projective complex curve. Formal-analytic surfaces provide a natural framework for arithmetic algebraization theorems, old and new. Formal-analytic arithmetic surfaces admit a rich geometry which parallels the geometry of complex analytic surfaces. Notably the dichotomy between pseudoconvexity and pseudoconcavity plays a central role in their geometry. Our study of formal-analytic arithmetic surfaces relies crucially on the use of real-valued invariants. Some of these are intersection-theoretic, in the spirit of Arakelov intersection theory. Some other invariants involve infinite-dimensional geometry of numbers. Relating our new intersection-theoretic invariants to more classical invariants of Arakelov geometry leads us to investigate a new invariant, the Archimedean overflow, attached to an analytic map from a pointed compact Riemann surface with boundary to a Riemann surface. It is related to the characteristic functions of Nevanlinna theory. Our results on the geometry of formal-analytic arithmetic surfaces admit applications to concrete problems of arithmetic geometry. Notably we generalize the arithmetic holonomicity theorem of Calegari-Dimitrov-Tang regarding the dimension of spaces of power series with integral coefficients satisfying some convergence conditions. We also establish an arithmetic counterpart of theorems of Lefschetz and Nori by providing a bound on the index, in the étale fundamental group of an arithmetic surface, of the closed subgroup generated by the étale fundamental groups of some arithmetic curve and of some compact Riemann surfaces mapping to the arithmetic surface.
