Packets in Grothendieck's Section Conjecture

Some implications between Grothendieck's anabelian conjectures

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…

Galois sections and $p$-adic period mappings

Let $K$ be a number field not containing a CM subfield. For any smooth projective curve $Y/K$ of genus $\geq2$, we prove that the image of the "Selmer" part of Grothendieck's section set inside the $K_v$-rational points $Y(K_v)$ is finite for every finite place $v$. This gives an unconditional verification of a prediction of Grothendieck's section conjecture. In the process of proving our main result, we also refine and extend the method of Lawrence and Venkatesh, with potential consequences for explicit computations.

Une remarque sur les courbes de Reichardt-Lind et de Schinzel

We prove that the arithmetic fundamental group of X admits no section over the absolute Galois group of Q when X is the Schinzel curve, thereby confirming in this example the prediction given by Grothendieck's section conjecture. ----- Nous démontrons que le groupe fondamental arithmétique de X n'admet pas de section au-dessus du groupe de Galois absolu de Q lorsque X est la courbe de Schinzel, confirmant ainsi sur cet exemple la prédiction donnée par la conjecture des sections de Grothendieck.

Good Sections of Arithmetic Fundamental Groups

In this paper we exhibit the notion of (uniformly) good sections of arithmetic fundamental groups. We introduce and investigate the problem of cuspidalisation of sections of arithmetic fundamental groups, its ultimate aim is to reduce the solution of the Grothendieck anabelian section conjecture to the solution of its birational version. We show that (uniformly) good sections of arithmetic fundamental groups of smooth, proper, and geometrically connected hyperbolic curves over slim (and regular) fields can be lifted to sections of cuspidally abelian absolute Galois groups. As an application we…

The Brauer-Manin obstruction for sections of the fundamental group

We establish Grothendieck's section conjecture for an open subset of the Reichardt-Lind curve, and introduce the notion of a Brauer-Manin obstruction for sections of the fundamental group extension.

The fundamental groupoid scheme and applications

We define a linear structure on Grothendieck's arithmetic fundamental group $π_1(X, x)$ of a scheme $X$ defined over a field $k$ of characteristic 0. It allows us to link the existence of sections of the Galois group ${\rm Gal}(\bar k/k)$ to $π_1(X, x)$ with the existence of a neutral fiber functor on the category which linearizes it. When applied to Grothendieck section conjecture, it allows us to find a $k$-structure on the universal covering, and $k$-rational pro-points at finite and infinite distance which lift given $k$-rational points. Changes as compared to the first version: (some) typ…

Arithmetic in the fundamental group of a p-adic curve: On the p-adic section conjecture for curves

We establish a valuative version of Grothendieck's section conjecture for curves over p-adic local fields. The image of every section is contained in the decomposition subgroup of a valuation which prolongs the p-adic valuation to the function field of the curve.

Topics in the Grothendieck conjecture for hyperbolic polycurves of dimension 2

In this paper, we study the anabelian geometry of hyperbolic polycurves of dimension 2 over sub-p-adic fields. In 1-dimensional case, Mochizuki proved the Hom version of the Grothendieck conjecture for hyperbolic curves over sub-p-adic fields and the pro-p version of this conjecture. In 2-dimensional case, a naive analogue of this conjecture does not hold for hyperbolic polycurves over general sub-p-adic fields. Moreover, the Isom version of the pro-p Grothendieck conjecture does not hold in general. We explain these two phenomena and prove the Hom version of the Grothendieck conjecture for hy…

On the section conjecture over function fields and finitely generated fields

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.

Chabauty-Kim, 有限降阶，以及局部几何截面的截面猜想

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。

$p$-adic 域上环面基本群的截面猜想

The section conjecture for the toric fundamental group over $p$-adic fields

本文在 $p$-adic 域上证明了环面基本群的 Grothendieck 截面猜想，并将 Selmer 截面重新解释为环面 Galois 截面。

Un critère d'épointage des sections $l$-adiques

The cuspidalization conjecture emerged as an approach of Grothendieck's famous section conjecture. We address a weak form of it by using a mild generalization of a theorem of Uwe Jannsen which describes exactly when the $l$-adic homology of an open curve is a pure Galois representation. We also give some concrete examples of modular curves for which the cuspidalization is possible at the $l$-adic level.

Lifting Galois sections along torsors

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…

Galois sections for abelianized fundamental groups

Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the étale fundamental group of $X$ onto the absolute Galois group of $k$ has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of $k$ but not over $k$. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the étale fundament…

On the "Section Conjecture" in anabelian geometry

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…

Invitation to higher local fields, Part II, section 7: Recovering higher global and local fields from Galois groups - an algebraic approach

A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an algebraic proof of the 0-dimensional case of Grothendieck's anabelian conjecture (proven by Pop), which says that finitely generated infinite fields are determined up to purely inseparable extensions by their absolute Galois groups. As a second application (which is a joint work with Fesenko) we analyze the arithmetic structure of fields with the same absolute …

Descent obstruction and fundamental exact sequence

A torsor under a k-group scheme G on a variety X over a number field k imposes a descent obstruction against the existence of rational points on X. We discuss the finite descent obstruction, that is for all such torsors under finite k-groups G, in view of a local-global interpolation property for sections of the fundamental group short exact sequence of X/k. There are applications to the Brauer-Manin obstruction, to the descent obstruction by torsors under linear groups, and to the birational version of Grothendieck's section conjecture over number fields. In particular, we obtain examples of …

Rational points of universal curves

In this paper we prove a version of Grothendieck's section conjecture for the restriction of the universal complete curve over M_{g,n}, g > 4, to the function field k(M_{g,n}) where k is, for example, a number field. In this version, the fundamental group of the closed fiber is replaced by its ell-adic unipotent completion when n > 1.

几何Batyrev-Manin-Peyre理论对于加法群等价紧缩的渐进行为

Geometric Batyrev-Manin-Peyre for equivariant compactifications of additive groups

本文研究了给定族在光滑射影曲线上的截面模空间的渐进行为，当通用纤维为有限维向量空间的等价紧缩时，证明了在合适的Grothendieck概形环中，这些模空间的类在某种归一化下收敛到非零有效元素，为Batyrev-Manin-Peyre猜想的动机版本提供了肯定答案。

On Profinite Hyperbolicity and Diophantine Geometry

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…

曲面丛与截面猜想

Surface bundles and the section conjecture

本文提出格罗滕迪克截面猜想的热带类比，构造两个伽罗瓦上同调类 o_1 和 o_2 作为截面与有理点的阻碍，并借助拓扑退化技巧在 p 进域和数域上给出大量满足截面猜想的曲线实例。

Arithmetic Orr invariants of absolute Galois groups

Based on the analogies between mapping class groups and absolute Galois groups, we introduce an arithmetic pro-$\ell$ analogue of Orr invariants for a Galois element associated with Galois action on étale fundamental groups of punctured projective lines. At the same time, we also introduce pro-$\ell$ Orr space as an arithmetic analogue of Orr space whose third homotopy group is a target group of Orr invariant. We then determine its rank as $\mathbb{Z}_{\ell}$-module following Igusa-Orr's computation. Moreover, we investigate its relation with Ellenberg's obstruction to $π_1$-sections associate…