Inter-universal Teichmüller Theory II: Hodge–Arakelov-Theoretic Evaluation
Inter-universal Teichmüller Theory II: Hodge–Arakelov-Theoretic Evaluation
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跨宇宙 Teichmüller 理论 II：Hodge-Arakelov 理论评估
DOI：
10.4171/prims/57-1-2
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发表时间：
2021
影响因子：
1.2
通讯作者：
S. Mochizuki
中科院分区：
文献类型：
--
作者：
S. Mochizuki
本文是四篇论文系列中的第二篇，我们研究了围绕 Hodge-Arakelov 理论评估的 Kummer 理论，即以作者在之前论文中建立的图式理论 Hodge-Arakelov 理论的风格进行评估，在 l 扭转点 [严格来说，通过合适的 2 扭转点移动] 处的 theta 函数 [l 次根的倒数]，对于 l ≥ 5一个质数。在该系列的第一篇论文中，我们研究了“传统方案理论的微型模型”，我们将其称为 θ±ellNF-Hodge 剧场，它们与某些称为初始 θ 数据的数据相关联，其中包括数域 F 上的椭圆曲线 EF 以及素数 l ≥ 5。这些 θ±ellNF-Hodge 剧场的底层 θ-Hodge 剧场通过以下方式彼此粘合： “θ-链接”，识别一个 θ ±ellNF-Hodge 剧场中 EF 不良还原素数处的 [l 次根的倒数] theta 函数，以及另一个 θ±ellNF-Hodge 剧场中 EF 不良还原素数处的 [2l 次根] q 参数。本文提出的理论允许人们构建这种“θ-链接”的某些新版本。其中一个新版本是 θ ×μ gaulink，它与 θ-link 类似，但涉及 l 扭转点处的 theta 值，而不是 theta 函数本身。构成 θ ×μ gau-link 的结构的一个重要方面是研究多半径特性，即“算术全纯结构”的特性，或者更具体地说，环/方案结构，源自一个 θ±ellNF-Hodge 剧院，可以以这样的方式表述，以便从与原始相关的另一个 θ±ellNF-Hodge 剧院的算术全纯结构的角度来看是有意义的θ±ellNF-Hodge 剧院通过[非方案理论！] θ ×μ gau-link。例如，作者在早期论文中研究的 étale theta 函数的某些刚性特性可以在本系列论文的理论背景下解释为多径向特性。 θ ×μ gau 链接基础结构的另一个重要方面是通过 θ ±ellNF-Hodge 剧院的 F ± l 对称性研究“共轭同步”。共轭同步是指某种同构系统——它不存在任何共轭不确定性！ — 在评估 theta 函数的各个 l 扭转点处的局部绝对伽罗瓦群的副本之间。共轭同步在围绕 l 扭转点处的 theta 函数评估的 Kummer 理论中发挥着重要作用，并应用于研究 θ ×μ gau 连杆的扭力性质 [即研究物体保持不变]。共轭同步的全局方面需要通过该系列第一篇论文中获得的结果来解决涉及调质尖端惯性群的有限共轭的某些技术细节。排版：AMS-TEX 1 2 望月新一
In the present paper, which is the second in a series of four papers, we study theKummer theory surrounding the Hodge-Arakelov-theoretic evaluation — i.e., evaluation in the style of the scheme-theoretic Hodge-Arakelov theory established by the author in previous papers — of the [reciprocal of the lth root of the] theta function at l-torsion points [strictly speaking, shifted by a suitable 2-torsion point], for l ≥ 5 a prime number. In the first paper of the series, we studied “miniature models of conventional scheme theory”, which we referred to as Θ±ellNF-Hodge theaters, that were associated to certain data, called initial Θ-data, that includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. The underlying Θ-Hodge theaters of these Θ±ellNF-Hodge theaters were glued to one another by means of “Θ-links”, that identify the [reciprocal of the l-th root of the] theta function at primes of bad reduction of EF in one Θ ±ellNF-Hodge theater with [2l-th roots of] the q-parameter at primes of bad reduction of EF in another Θ±ellNF-Hodge theater. The theory developed in the present paper allows one to construct certain new versions of this “Θ-link”. One such new version is the Θ ×μ gaulink, which is similar to the Θ-link, but involves the theta values at l-torsion points, rather than the theta function itself. One important aspect of the constructions that underlie the Θ ×μ gau-link is the study of multiradiality properties, i.e., properties of the “arithmetic holomorphic structure” — or, more concretely, the ring/scheme structure — arising from one Θ±ellNF-Hodge theater that may be formulated in such a way as to make sense from the point of the arithmetic holomorphic structure of another Θ±ellNF-Hodge theater which is related to the original Θ±ellNF-Hodge theater by means of the [non-scheme-theoretic!] Θ ×μ gau-link. For instance, certain of the various rigidity properties of the étale theta function studied in an earlier paper by the author may be intepreted as multiradiality properties in the context of the theory of the present series of papers. Another important aspect of the constructions that underlie the Θ ×μ gau-link is the study of “conjugate synchronization” via the F ± l -symmetry of a Θ ±ellNF-Hodge theater. Conjugate synchronization refers to a certain system of isomorphisms — which are free of any conjugacy indeterminacies! — between copies of local absolute Galois groups at the various l-torsion points at which the theta function is evaluated. Conjugate synchronization plays an important role in the Kummer theory surrounding the evaluation of the theta function at l-torsion points and is applied in the study of coricity properties of [i.e., the study of objects left invariant by] the Θ ×μ gau-link. Global aspects of conjugate synchronization require the resolution, via results obtained in the first paper of the series, of certain technicalities involving profinite conjugates of tempered cuspidal inertia groups. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI
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发表时间：
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发表时间：
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