Mohamed Saidi, Akio Tamagawa
In this paper, we prove a prime-to-p version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields of characteristic p>0, whose original (full profinite) version was proved by Tamagawa in the affine case and by Mochizuki in the proper case.
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We prove that a certain class of open homomorphisms between Galois groups of function fields of curves over finite fields arise from embeddings between the function fields.
This paper is around the topics I discussed in the lecture I gave at the Isaac Newton Institute in Cambridge, July 2009, in the Introductory Workshop. This paper can be read as a companion to my paper [Saï di], where detailed proofs can be found.
In this paper we prove a refined version of Uchida's theorem on isomorphisms between absolute Galois groups of global fields in positive characteristics, where one "ignores" the information provided by a "small" set of primes.
Grothendieck gave two forms of his "main conjecture of anabelian geometry", i.e. the section conjecture and the hom conjecture. He stated that these two forms are equivalent and that if they hold for hyperbolic curves then they hold for elementary anabelian varieties too. We state a stronger form of Grothendieck's conjecture (equivalent in the case of curves) and prove that Grothendieck's statements hold for our form of the conjecture. We work with DM stacks, rather than schemes. If $X$ is a DM stack over a field $k\subseteq\mathbb{C}$ finitely generated over $\mathbb{Q}$, we prove that whethe…
In this paper we prove a refined version of a theorem by Tamagawa and Mochizuki on isomorphisms between (tame) arithmetic fundamental groups of hyperbolic curves over finite fields, where one "ignores" the information provided by a "small" set of primes.
My view on and experience with Grothendieck's anabelian geometry
本文基于作者在查普曼大学Grothendieck会议的报告，阐述了其对Grothendieck阿贝尔几何的个人观点与经验，并非该领域的综述。
In this paper we will prove that there exists a covariant functor, called algebraic anabelian functor, from the category of algebraic schemes over a given field to the category of outer homomorphism sets of groups. The algebraic anabelian functor, given in a canonical manner, is full and faithful. It reformulates the anabelian geometry over a field. As an application of the anabelian functor, we will also give a proof of the section conjecture of Grothendieck for the case of algebraic schemes.
An investigation of morphisms that coincide topologically is used to generalize to all characteristics and partly reprove Tamagawa's theorem on the Grothendieck conjecture in anabelian geometry for affine hyperbolic curves. The theorem now deals with π^{tame} of curves over a finitely generated field and its effect on the set of isomorphisms. Universal homeomorphisms are formally inverted.
Anabelian properties of infinite algebraic extensions of finite fields
本文研究有限域无限代数扩张上的远阿贝尔几何，将 Tamagawa 与 Sarashina 关于双曲曲线的结果推广至此类域，并用绝对 Galois 群刻画其（环向）Kummer-faithful 的充要条件。
Anabelian geometry with etale homotopy types generalizes in a natural way classical anabelian geometry with etale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field k which is finitely generated over Q has a fundamental system of (affine) anabelian Zariski-neighbourhoods. This was predicted by Grothendieck in his letter to Faltings.
In the present paper, we study a new kind of anabelian phenomenon concerning the smooth pointed stable curves in positive characteristic. It shows that the topological structures of moduli spaces of curves can be understood from the viewpoint of anabelian geometry. We formulate some new anabelian-geometric conjectures relating the tame fundamental groups of curves over algebraically closed fields of characteristic $p>0$ to the moduli spaces of curves. These conjectures are generalized versions of the weak Isom-version of the Grothendieck conjecture for curves over algebraically closed fields o…
The geometrically m-step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields
本文在非交换几何中针对素域上有限生成域上的仿射双曲曲线，通过先证有限域情形、再证Oda-Tamagawa准则的几何m步可解版本，最终证明了（弱与强双非交换的）几何m步可解Grothendieck猜想。
Resolution of non-singularities and the absolute anabelian conjecture
本文针对p进局部域上双曲曲线的绝对远阿贝尔猜想，将S. Mochizuki针对准Belyi型曲线的证明结果推广到包含双曲Mumford曲线在内的更广泛曲线族。
In this survey, we introduce the three theorems about the m-step solvable Grothendieck conjecture in anabelian geometry of hyperbolic curves by H. Nakamura, S. Mochizuki, and the author. We also give sketches of the proofs of these theorems.
Etale descent obstruction and anabelian geometry of curves over finite fields
本文通过建立平展基本群之间的态射与全局函数域上曲线的局部常值阿代尔点之间的双射，将Grothendieck的阿纳贝尔猜想与Sutherland和第二个作者的最新猜想联系起来，为阿纳贝尔猜想提供了进一步证据。
Let X be a smooth projective curve of genus >1 over a field K which is finitely generated over the rationals. The section conjecture in Grothendieck's anabelian geometry says that the sections of the canonical projection from the arithmetic fundamental group of X onto the absolute Galois group of K are (up to conjugation) in one-to-one correspondence with K-rational points of X. The birational variant conjectures a similar correspondence where the fundamental group is replaced by the absolute Galois group of the function field K(X). The present paper proves the birational section conjecture fo…
A Note on Hodge theoretic anabelian geometry
本文提出一个基于Hodge理论的anabelian猜想，通过非阿贝尔Hodge理论中的ℂ×作用替代Galois作用，并证明了双曲曲线和复双曲流形的类比定理。
We study the section conjecture of anabelian geometry and the sufficiency of the finite descent obstruction to the Hasse principle for the moduli spaces of principally polarized abelian varieties and of curves over number fields. For the former we show that the section conjecture fails and the finite descent obstruction holds for a general class of adelic points, assuming several well-known conjectures. This is done by relating the problem to a local-global principle for Galois representations. For the latter, we prove some partial results that indicate that the finite descent obstruction suff…
Anabelian Geometry in Families
本文在次p进域上有限型正规基概型S上，研究双曲曲线族的阿贝尔几何性质，证明其相对于优态射在光滑S概型中具有阿贝尔性，拓展了阿贝尔几何的研究范围。
The geometrically m-step solvable Grothendieck conjecture for genus 0 curves over finitely generated fields
研究远阿贝尔几何中的几何 m 步可解 Grothendieck 猜想，证明任意特征素域的有限生成扩域上零亏格曲线满足几何 3 步可解情形。
Center-freeness of finite-step solvable groups arising from anabelian geometry
研究双曲曲线的最大m步可解商的无中心性，证明其无挠且无中心，进而推导出m步可解Grothendieck猜想的刚性性质。
We introduce the category of structures and interpretations which allows us to discuss some issues of Grothendieck's anabelian geometry in model-theory terms. Our main result is a formulation in terms of pure stability theory of a problem closely related to Grothendieck's section conjecture
We finish the proof of the conjecture of F. Bogomolov and F. Pop: Let $F_{1}$ and $F_{2}$ be fields finitely-generated and of transcendence degree $\geq 2$ over $k_{1}$ and $k_{2}$, respectively, where $k_{1}$ is either $\bar{\mathbb{Q}}$ or $\bar{\mathbb{F}}_{p}$, and $k_{2}$ is algebraically closed. We denote by $G_{F_1}$ and $G_{F_2}$ their respective absolute Galois groups. Then the canonical map $φ_{F_{1}, F_{2}}: \Isom^i(F_1, F_2)\rightarrow \Isom^{\Out}_{\cont}(G_{F_2}, G_{F_1})$ from the isomorphisms, up to Frobenius twists, of the inseparable closures of $F_1$ and $F_2$ to continuous …
In this work we bring to light some anabelian behaviours of analytic curves in the setting of Berkovich geometry. We show more precisely that the knowledge of the tempered fundamental group of some curves that we call analytically anabelian determines their analytic skeletons as graphs. The tempered fundamental group of a Berkovich space, introduced by André, enabled Mochizuki to prove the first result of anabelian geometry in Berkovich geometry concerning analytifications of algebraic hyperbolic curves over $\overline{\mathbb{Q}}_p$. To that end, Mochizuki developed the categorical language o…
We study hyperbolic curves and their Jacobians over finite fields in the context of anabelian geometry.
Profinite complexes of curves, their automorphisms and anabelian properties of moduli stacks of curves
本文证明了定义在sub-$p$-adic域上的曲线模栈的有限连通étale覆盖满足Grothendieck猜想意义下的“几乎”anabelian性质，建立了其自同构群与几何基本群外部自同构子群之间的自然同构，将双曲曲线的anabelian性质部分扩展至曲线模空间。
Rational points in the boundary of a hyperbolic curve over a field with sufficiently nontrivial Kummer theory are the source for an abundance of sections of the fundamental group exact sequence. We follow and refine Nakamura's approach towards these boundary sections. For example, we obtain a weak anabelian theorem for hyperbolic genus 0 curves over quite general fields including for example the maximal abelian extension of the rational numbers.
We investigate sections of arithmetic fundamental groups of hyperbolic curves over function fields. As a consequence we prove that the anabelian section conjecture of Grothendieck holds over all finitely generated fields over $\Bbb Q$ if it holds over all number fields, under the condition of finiteness (of the $\ell$-primary parts) of certain Shafarevich-Tate groups. We also prove that if the section conjecture holds over all number fields then it holds over all finitely generated fields for curves which are defined over a number field.
On Grothendieck's section conjecture for curves of index $1$
本文证明了具有非循环 $p$-群忠实作用的双曲曲线存在满足 Grothendieck 截面猜想的指标为 $1$ 的扭曲形式，并证明了在 $\mathbb{Q}$ 上有限生成域上的双曲曲线经有限扩张与有限平展覆盖后可满足该猜想。
Reconstruction of schemes from their étale topoi
本文对任意特征的无限域证明了Grothendieck关于将 $k$-概形映射至其étale topos构成完全忠实函子的猜想，实现了概形的重构并推广了Voevodsky的工作。
The metabelian Grothendieck conjecture for genus zero curves over finitely generated fields
研究有限生成域上零亏格曲线的亚交换格罗滕迪克猜想，通过证明两条双曲零亏格曲线在基域上同构（正特征下经弗罗贝尼乌斯扭转）的充要条件是其几何极大亚交换驯基本群在基域绝对伽罗瓦群上同构来得出结论。
The étale topos reconstructs varieties over sub-p-adic fields
本文证明子p进域上有限型概形及其相对版本可由étale拓扑空间在泛同胚意义下重构，推广了Voevodsky结果并部分验证Grothendieck猜想。
A birationally liftable Galois section s of a hyperbolic curve X/k over a number field k yields an adelic point x(s) in the smooth completion of X. We show that x(s) is X-integral outside a set of places of Dirichlet density 0, or s is cuspidal. The proof relies on $GL_2(F_\ell)$-quotients of $π_1(U)$ for some open U of X. If k is totally real or imaginary quadratic, we prove that all birationally adelic, non-cuspidal Galois sections come from rational points as predicted by the section conjecture of anabelian geometry. As an aside we also obtain a strong approximation result for rational poin…
In this note, we explore the notion of hyperbolicity of topologically finitely generated profinite groups. Some applications to diophantine geometry are suggested and we try to reformulate certain problems in diophantine geometry in terms of hyperbolic profinite groups. Then, we introduce many occasions in which Galois groups are free profinite and try to explore implications of this condition in the world of diophantine geometry. In particular, we prove that, Grothendieck's "section conjecture" plus Shafarevich's "freeness conjecture" imply that hyperbolic curves have infinitely many solution…
A. Vistoli observed that, if Grothendieck's section conjecture is true and $X$ is a smooth hyperbolic curve over a field finitely generated over $\mathbb{Q}$, then $\underlineπ_{1}(X)$ should somehow have essential dimension $1$. We prove that an infinite, pro-finite étale group scheme always has infinite essential dimension. We introduce a variant of essential dimension, the fce dimension $\operatorname{fced} G$ of a pro-finite group scheme $G$, which naturally coincides with $\operatorname{ed} G$ if $G$ is finite but has a better behaviour in the pro-finite case. Grothendieck's section conje…
Quadric bundles and hyperbolic equivalence
本文定义二次丛与向量丛二次型的双曲等价，刻画其 Brauer 数据、双有理与 Grothendieck 类等不变量，并给出射影空间上的判定准则及 VLC 丛化简。
Chabauty--Kim, finite descent, and the Section Conjecture for locally geometric sections
本文研究了Chabauty-Kim猜想与局部几何截面猜想，证明了在满足Kim猜想的条件下，X/Q满足截面猜想，并推广到超几何曲线的S-整点。
The cuspidalization conjecture, which is a consequence of Grothendieck's section conjecture, asserts that for any smooth hyperbolic curve $X$ over a finitely generated field $k$ of characteristic $0$ and any non empty Zariski open $U \subset X$, every section of $π_1 (X, \bar x) \to \mathrm{Gal}_k$ lifts to a section of $π_1 (U,\bar x) \to \mathrm{Gal}_k$. We consider in this article the problem of lifting Galois sections to the intermediate quotient $ π_1^{cc}(U)$ introduced by Mochizuki. We show that when $k = \mathbb Q$ and $D=X\setminus U$ is an union of torsion sub-packets every Galois se…
A motivic Weil height machine for curves
本文提出了一种针对曲线的动机Weil高度机器，用于研究双曲曲线的有理点集与动机增广集的关系，并证明了特定曲线的动机增广的有限性结果。
Bitangents to plane quartics via tropical geometry: rationality, $\mathbb{A}^1$-enumeration, and real signed count
本文利用热带几何研究平面四次曲线双切线的算术枚举问题，给出有理性障碍和GW重数的热带刻画，并证明多数热带双切线类对$\mathbb{A}^1$-枚举贡献$2\mathbb{H}$。
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