Section Conjecture Research Articles

Recovering p$p$‐adic valuations from pro‐p$p$ Galois groups

AbstractLet be a field with , where denotes the maximal pro‐2 quotient of the absolute Galois group of a field . We prove that then admits a (non‐trivial) valuation which is 2‐henselian and has residue field . Furthermore, is a minimal positive element in the value group and . This forms the first positive result on a more general conjecture about recovering ‐adic valuations from pro‐ Galois groups which we formulate precisely. As an application, we show how this result can be used to easily obtain number‐theoretic information, by giving an independent proof of a strong version of the birational section conjecture for smooth, complete curves over , as well as an analogue for varieties.

TOPICS SURROUNDING THE COMBINATORIAL ANABELIAN GEOMETRY OF HYPERBOLIC CURVES IV: DISCRETENESS AND SECTIONS

Abstract Let $\Sigma $ be a nonempty subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one. In the present paper, we continue our study of the pro- $\Sigma $ fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields in which the primes of $\Sigma $ are invertible. The present paper focuses on the topic of comparison between the theory developed in earlier papers concerning pro- $\Sigma $ fundamental groups and various discrete versions of this theory. We begin by developing a theory concerning certain combinatorial analogues of the section conjecture and Grothendieck conjecture. This portion of the theory is purely combinatorial and essentially follows from a result concerning the existence of fixed points of actions of finite groups on finite graphs (satisfying certain conditions). We then examine various applications of this purely combinatorial theory to scheme theory. Next, we verify various results in the theory of discrete fundamental groups of hyperbolic topological surfaces to the effect that various properties of (discrete) subgroups of such groups hold if and only if analogous properties hold for the closures of these subgroups in the profinite completions of the discrete fundamental groups under consideration. These results make possible a fairly straightforward translation, into discrete versions, of pro- $\Sigma $ results obtained in previous papers by the authors. Finally, we discuss a construction that was considered previously by M. Boggi in the discrete case from the point of view of the present paper.

On the birational section conjecture with strong birationality assumptions

Let X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document} be a curve over a field k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k$\\end{document} finitely generated over ℚ and t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$t$\\end{document} an indeterminate. We prove that, if s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$s$\\end{document} is a section of π1(X)→Gal(k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\pi _{1}(X)\ o \\operatorname{Gal}(k)$\\end{document} such that the base change sk(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$s_{k(t)}$\\end{document} is birationally liftable, then s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$s$\\end{document} comes from geometry. As a consequence we prove that the section conjecture is equivalent to the cuspidalization of all sections over all finitely generated fields.

Rational motivic path spaces and Kim’s relative unipotent section conjecture

We develop the foundations of commutative algebra objects in the category of motives, which we call “motivic dga’s.” Works of White and Cisinski and Déglise provide us with a suitable model structure. This enables us to reconstruct the unipotent fundamental group of a pointed scheme from the associated augmented motivic dga and provides us with a factorization of Kim’s relative unipotent section conjecture into several smaller conjectures with a homotopical flavor.

Surface bundles and the section conjecture

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of this conjecture. In so doing we produce many examples of curves satisfying the section conjecture over fields of geometric interest, and then over p-adic fields and number fields via a Chebotarev argument. We construct two Galois cohomology classes o_1 and o_2, which obstruct the existence of pi_1-sections and hence of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli space of curves M_g studied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes obstruct sections. We then use these constructions to produce curves over p-adic fields and number fields where each class obstructs pi_1-sections and hence rational points. Among our geometric results are a new proof of the section conjecture for the generic curve of genus g>2, and a proof of the section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the existence of a section is genuinely non-abelian).

Arithmetic Orr Invariants of Absolute Galois Groups

Abstract 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 $\pi _1$-sections associated with lower central series filtration in the context of Grothendieck’s section conjecture.

The section conjecture over large algebraic extensions of finitely generated fields

AbstractLet K be a finitely generated extension of its prime field and let be an integer. We prove the injectivity part of the section conjecture of Grothendieck for almost all and for all smooth geometrically integral projective curves of genus ≥1 over the field .

Essential dimension and pro-finite group schemes

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 $\underline{\pi}_{1}(X)$ should somehow have essential dimension $1$. We prove that an infinite, pro-finite \'etale 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 conjecture implies $\operatorname{fced}\underline{\pi}_{1}(X)=\dim X=1$ for $X$ as above. We prove that, if $A$ is an abelian variety over a field finitely generated over $\mathbb{Q}$, then $\operatorname{fced}\underline{\pi}_{1}(A)=\operatorname{fced} TA=\dim A$.

On the section conjecture and Brauer\u2013Severi varieties

J. Stix proved that a curve of positive genus over mathbb {Q} which maps to a non-trivial Brauer–Severi variety satisfies the section conjecture. We prove that, if X is a curve of positive genus over a number field k and the Weil restriction R_{k/mathbb {Q}}X admits a rational map to a non-trivial Brauer–Severi variety, then X satisfies the section conjecture. As a consequence, if X maps to a Brauer–Severi variety P such that the corestriction {text {cor}}_{k/mathbb {Q}}([P])in {text {Br}}(mathbb {Q}) is non-trivial, then X satisfies the section conjecture.

On the birational section conjecture over finitely generated fields

We investigate the birational section conjecture for curves over function fields of characteristic zero and prove that the conjecture holds over finitely generated fields over Q if it holds over number fields.

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 whether $X$ satisfies the conjecture or not depends only on $X_{\mathbb{C}}$. We prove that the section conjecture for hyperbolic orbicurves stated by Borne and Emsalem follows from the conjecture for hyperbolic curves.

Morphisms of Rational Motivic Homotopy Types

We investigate several interrelated foundational questions pertaining to the study of motivic dga’s of Dan-Cohen and Schlank (Rational motivic path spaces and Kim’s relative unipotent section conjecture. arXiv:1703.10776 ) and Iwanari (Motivic rational homotopy type. arXiv:1707.04070 ). In particular, we note that morphisms of motivic dga’s can reasonably be thought of as a nonabelian analog of motivic cohomology. Just as abelian motivic cohomology is a homotopy group of a spectrum coming from K-theory, the space of morphisms of motivic dga’s is a certain limit of such spectra; we give an explicit formula for this limit—a possible first step towards explicit computations or dimension bounds. We also consider commutative comonoids in Chow motives, which we call “motivic Chow coalgebras”. We discuss the relationship between motivic Chow coalgebras and motivic dga’s of smooth proper schemes. As a small first application of our results, we show that among schemes which are finite etale over a number field, morphisms of associated motivic dga’s are no different than morphisms of schemes. This may be regarded as a small consequence of a plausible generalization of Kim’s relative unipotent section conjecture, hence as an ounce of evidence for the latter.

Duality for Commutative Group Stacks

Abstract We study in this article the dual of a (strictly) commutative group stack $G$ and give some applications. Using the Picard functor and the Picard stack of $G$, we first give some sufficient conditions for $G$ to be dualizable. Then, for an algebraic stack $X$ with suitable assumptions, we define an Albanese morphism $a_X: X\longrightarrow A^1(X)$ where $A^1(X)$ is a torsor under the dual commutative group stack $A^0(X)$ of $\textrm{Pic}_{X/S}$. We prove that $a_X$ satisfies a natural universal property. We give two applications of our Albanese morphism. On the one hand, we give a geometric description of the elementary obstruction and of universal torsors (standard tools in the study of rational varieties over number fields). On the other hand, we give some examples of algebraic stacks that satisfy Grothendieck’s section conjecture.

Rational points of universal curves in positive characteristics

Let K K be the function field of the moduli stack M g , n / F p {\mathcal M}_{g,n/{\mathbb F}_p} of curves over Spec F p \text {Spec}{\mathbb F}_p , and let C / K C/K be the restriction of the universal curve to Spec K \text {Spec} K . We show that if g ≥ 3 g\geq 3 , then the only K K -rational points of C C are its n n tautological points. Furthermore, we show that if g ≥ 4 g\geq 4 and n = 0 n=0 , then Grothendieck’s section conjecture holds for C C over K K . This extends Hain’s work in characteristic 0 0 to positive characteristics.

On the sections of universal hyperelliptic curves

In this paper, we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves CHg,n/k→Hg,n/k, when g≥3 and the image of the ℓ-adic cyclotomic character Gk→Z× is infinite. Furthermore, we will study the nonabelian phenomena associated to the universal hyperelliptic curves. For example, we will show that the section conjecture holds for CHg/k→Hg/k and the unipotent analogue of the conjecture holds for the pointed cases. This work is an extension of Hain's original work on the rational points of universal curves to the hyperelliptic case.

A non-abelian conjecture of Tate–Shafarevich type for hyperbolic curves

Let X denote a hyperbolic curve over $$\mathbb {Q}$$ and let p denote a prime of good reduction. The third author’s approach to integral points, introduced in Kim (Invent Math 161:629–656, 2005; Publ Res Inst Math Sci 45:89–133, 2009), endows $$X({\mathbb {Z}_p})$$ with a nested sequence of subsets $$X({\mathbb {Z}_p})_n$$ which contain $$X(\mathbb {Z})$$ . These sets have been computed in a range of special cases (Balakrishnan et al., J Am Math Soc 24:281–291, 2011; Dan-Cohen and Wewers, Proc Lond Math Soc 110:133–171, 2015; Dan-Cohen and Wewers, Int Math Res Not IMRN 17:5291–5354, 2016; Kim, J Am Math Soc 23:725–747, 2010); there is good reason to believe them to be practically computable in general. In 2012, the third author announced the conjecture that for n sufficiently large, $$X(\mathbb {Z}) = X({\mathbb {Z}_p})_n$$ . This conjecture may be seen as a sort of compromise between the abelian confines of the BSD conjecture and the profinite world of the Grothendieck section conjecture. After stating the conjecture and explaining its relationship to these other conjectures, we explore a range of special cases in which the new conjecture can be verified.

The metabelian birational -adic section conjecture for varieties

We generalize the $\mathbb{Z}/p$metabelian birational $p$-adic section conjecture for curves, as introduced and proved in Pop [On the birational$p$-adic section conjecture, Compos. Math. 146 (2010), 621–637], to all complete smooth varieties, provided $p>2$. The condition $p>2$ seems to be of technical nature only, and might be removable.

On the Section Conjecture over Function Fields and Finitely Generated Fields

We investigate sections of the 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 \mathbb Q if it holds over all number fields, under the condition of finiteness (of the \mathcal l -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.

Le groupe fondamental étale d’un espace homogène d’un groupe algébrique linéaire

Let X be a homogeneous space of a connected linear algebraic group G defined over an algebraic closed field k of characteristic exponent p. Let \(x\in X(k)\). We denote by H the stabilizer of x in G and we assumed that H is connected or abelian. In this text, we compute explicitely the prime-to-p-part of the etale fundamental group \(\pi _1^{\acute{\mathrm{e}}\mathrm{t}}(X,x)\) in terms of the character groups of G and H. As an application, we prove a variant of the section conjecture for homogeneous spaces.

Anabelian geometry and descent obstructions on moduli spaces

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 suffices. We also show how this sufficiency implies the same for all hyperbolic curves.