Etale Fundamental Group Research Articles

Etale descent obstruction and anabelian geometry of curves over finite fields

Let $C$ and $D$ be smooth, proper and geometrically integral curves over a finite field $F$. Any morphism from $D$ to $C$ induces a morphism of their \'etale fundamental groups. The anabelian philosophy proposed by Grothendieck suggests that, when $C$ has genus at least $2$, all open homomorphisms between the \'etale fundamental groups should arise in this way from a nonconstant morphism of curves. We relate this expectation to the arithmetic of the curve $C_K$ over the global function field $K = F(D)$. Specifically, we show that there is a bijection between the set of conjugacy classes of well-behaved morphism of fundamental groups and locally constant adelic points of $C_K$ that survive \'etale descent. We use this to provide further evidence for the anabelian conjecture by relating it to another recent conjecture by Sutherland and the second author.

Non-abelian cohomology of universal curves in positive characteristic

In this paper, we will compute the non-abelian cohomology of the universal complete curve in positive characteristic. This extends Hain’s result on the non-abelian cohomology of generic curves in characteristic zero to positive characteristics. Furthermore, we will prove that the exact sequence of etale fundamental groups of the universal n n -punctured curve in positive characteristic does not split.

An obstruction to lifting to characteristic 0

We introduce a new obstruction to lifting smooth proper varieties in characteristic $p>0$ to characteristic $0$. It is based on Grothendieck's specialization homomorphism and the resulting discrete finiteness properties of etale fundamental groups.

Group-theoretic Johnson classes and a non-hyperelliptic curve with torsion Ceresa class

Let l be a prime and G a pro-l group with torsion-free abelianization. We produce group-theoretic analogues of the Johnson/Morita cocycle for G -- in the case of surface groups, these cocycles appear to refine existing constructions when l=2. We apply this to the pro-l etale fundamental groups of smooth curves to obtain Galois-cohomological analogues, and discuss their relationship to work of Hain and Matsumoto in the case the curve is proper. We analyze many of the fundamental properties of these classes and use them to give an example of a non-hyperelliptic curve whose Ceresa class has torsion image under the l-adic Abel-Jacobi map.

Extension of torsors and prime to p fundamental group scheme

Let $R$ be a discrete valuation ring with fraction field $K$. Let $X$ be a proper and faithfully flat $R$-scheme, endowed with a section $x \in X(R)$, with connected and reduced generic fibre $X_{\eta}$. Let $f: Y \rightarrow X_{\eta}$ be a finite Nori-reduced $G$-torsor. In this paper we provide a useful criterion to extend $f: Y \rightarrow X_{\eta}$ to a torsor over $X$. Furthermore in the particular situation where $R$ is a complete discrete valuation ring of residue characteristic $p>0$ and $X\to \text{Spec}(R)$ is smooth we apply our criterion to prove that the natural morphism $\psi^{(p')}: \pi(X_{\eta},x_{\eta})^{(p')}\to \pi(X,x)_{\eta}^{(p')}$ between the prime-to-$p$ fundamental group scheme of $X_{\eta}$ and the generic fibre of the prime-to-$p$ fundamental group scheme of $X$ is an isomorphism. This generalizes a well known result for the \'etale fundamental group. The methods used are purely tannakian.

Density of Arithmetic Representations of Function Fields

We propose a conjecture on the density of arithmetic points in the deformation space of representations of the \'etale fundamental group in positive characteristic. This? conjecture has applications to \'etale cohomology theory, for example it implies a Hard Lefschetz conjecture. We prove the density conjecture in tame degree two for the curve $\mathbb{P}^1\setminus \{0,1,\infty\}$. v2: very small typos corrected.v3: final. Publication in Epiga.

Projective crystalline representations of étale fundamental groups and twisted periodic Higgs–de Rham flow

This paper contains three new results. {\bf 1}.We introduce new notions of projective crystalline representations and twisted periodic Higgs-de Rham flows. These new notions generalize crystalline representations of etale fundamental groups introduced in [7,10] and periodic Higgs-de Rham flows introduced in [19]. We establish an equivalence between the categories of projective crystalline representations and twisted periodic Higgs-de Rham flows via the category of twisted Fontaine-Faltings module which is also introduced in this paper. {\bf 2.}We study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometrically absolutely irreducible crystalline representation. {\bf 3.} We investigate the dynamic of self-maps induced by the Higgs-de Rham flow on the moduli spaces of rank-2 stable Higgs bundles of degree 1 on $\mathbb{P}^1$ with logarithmic structure on marked points $D:=\{x_1,\,...,x_n\}$ for $n\geq 4$ and construct infinitely many geometrically absolutely irreducible $\mathrm{PGL_2}(\mathbb Z_p^{\mathrm{ur}})$-crystalline representations of $\pi_1^\text{et}(\mathbb{P}^1_{\mathbb{Q}_p^\text{ur}}\setminus D)$. We find an explicit formula of the self-map for the case $\{0,\,1,\,\infty,\,\lambda\}$ and conjecture that a Higgs bundle is periodic if and only if the zero of the Higgs field is the image of a torsion point in the associated elliptic curve $\mathcal{C}_\lambda$ defined by $ y^2=x(x-1)(x-\lambda)$ with the order coprime to $p$.

On the Geometric Subgroups of the Étale Fundamental Groups of Varieties over Real Closed Fields

In the present paper, we establish a “group-theoretic” algorithm for reconstructing, from the etale fundamental group of a suitable proper normal variety over a real closed field, the geometric subgroup of the etale fundamental group of the proper normal variety.

Sur l'existence du sch\'ema en groupes fondametal

Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite type and $x\in X(S)$ a section. The aim of the present paper is to establish the existence of the fundamental group scheme of $X$, when $X$ has reduced fibers or when $X$ is normal. We also prove the existence of a group scheme, that we will call the quasi-finite fundamental group scheme of $X$ at $x$, which classifies all the quasi-finite torsors over $X$, pointed over $x$. We define Galois torsors, which play in this context a role similar to the one of Galois covers in the theory of \'etale fundamental group.

On étale fundamental groups of formal fibres of $p$-adic curves

We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) etale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric etale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-)prime-to-$p$ free of finite computable rank.

Wonderful compactification of character varieties

Using the wonderful compactification of a semisimple adjoint affine algebraic group G defined over an algebraically closed field k of arbitrary characteristic, we construct a natural compactification Y of the G-character variety of any finitely generated group F. When F is a free group, we show that this compactification is always simply connected with respect to the \'etale fundamental group, and when k=C it is also topologically simply connected. For other groups F, we describe conditions for the compactification of the moduli space to be simply connected and give examples when these conditions are satisfied, including closed surface groups and free abelian groups when G=PGL(n,C). Additionally, when F is a free group we identify the boundary divisors of Y in terms of previously studied moduli spaces, and we construct a family of Poisson structures on Y and its boundary divisors arising from Belavin-Drinfeld splittings of the double of the Lie algebra of G. In the appendix, authored by Sam Evens and Arlo Caine, we explain how to put a Poisson structure on a quotient of a Poisson algebraic variety by the action of a reductive Poisson algebraic group.

Reconstruction of one-punctured elliptic curves in positive characteristic by their geometric fundamental groups

Our main question is whether the isomorphism class as a scheme of a curve X over an algebraic closure of a finite field can be reconstructed by the etale fundamental group of X. Tamagawa answered this question affirmatively when the genus of X is 0. In this paper, we will discuss the genus 1 case, and prove a similar result when the genus of X is 1, the cardinality of cusps is 1, and the characteristic of X is not equal to 2.

Ramification theory and formal orbifolds in arbitrary dimension

Formal orbifolds are defined in higher dimension to study wild ramification. Their etale fundamental groups are also defined. It is shown that the fundamental groups of formal orbifolds have certain finiteness property and it is also shown that they can be used to approximate the etale fundamental groups of normal varieties. Etale site on formal orbifolds are also defined. This framework allows one to study wild ramification in an organized way. Brylinski–Kato filtration, Lefschetz theorem for fundamental groups and l-adic sheaves in these contexts are also studied.

The action of the etale fundamental group scheme on the connected component of the essentially finite one

We follow the pattern in a recent paper of Otabe [Ota15] to define an action of the \'etale fundamental group scheme $\pi^\text{et}(X)$ on the local component of the essentially finite fundamental group scheme $\pi^{\mathrm{EF}}(X)$ of Nori. We show that the associated representation is faithful when $X$ is a curve of genus $\geq 2$.

Degeneration of period matrices of stable curves

In the present paper, we study the extent to which linear combinations of period matrices arising from stable curves are degenerate (i.e., as bilinear forms). We give a criterion to determine whether a stable curve admits such a degenerate linear combination of period matrices. In particular, this criterion can be interpreted as a certain analogue of the weight-monodromy conjecture for non-degenerate elements of pro-$\ell$ log etale fundamental groups of certain log points associated to the log stack $\overline{\mathcal{M}}_{g}^{log}$.

Convergent isocrystals on simply connected varieties

It is conjectured by de Jong that, if X is a connected smooth projective variety over an algebraically closed eld k of characteristic p > 0 with trivial etale fundamental group, any isocrystal on X is constant. We prove this conjecture under certain additional assumptions.

Indecomposability of various profinite groups arising from hyperbolic curves

In this paper, we prove that the etale fundamental group of a hyperbolic curve over an arithmetic field [e.g., a finite extension field of Q or Qp] or an algebraically closed field is indecomposable [i.e., cannot be decomposed into the direct product of nontrivial profinite groups]. Moreover, in the case of characteristic zero, we also prove that the etale fundamental group of the configuration space of a curve of the above type is indecomposable. Finally, we consider the topic of indecomposability in the context of the comparison of the absolute Galois group of Q with the Grothendieck-Teichmuller group GT and pose the question: Is GT indecomposable? We give an affirmative answer to a pro-l version of this question

The Grothendieck conjecture for the moduli spaces of hyperbolic curves of genus one

We study the Grothendieck conjecture for the moduli spaces of hyperbolic curves of genus one. A consequence of the main results is that the isomorphism class of a certain moduli space of hyperbolic curves of genus one over a sub-$p$-adic field is completely determined by the isomorphism class of the etale fundamental group of the moduli space over the absolute Galois group of the sub-$p$-adic field. We also prove related results in absolute anabelian geometry.

GALOIS CLOSURE DATA FOR EXTENSIONS OF RINGS

To generalize the notion of Galois closure for separable field extensions, we devise a notion of G-closure for algebras of commutative rings R → A, where A is locally free of rank n as an R-module and G is a subgroup of S n . A G-closure datum for A over R is an R-algebra homomorphism φ : (A ⨂n ) G → R satisfying certain properties, and we associate to a closure datum φ a closure algebra A ⨂n $$ {\otimes}_{{\left({A}^{\otimes n}\right)}^G} $$ R. This construction reproduces the normal closure of a finite separable field extension if G is the corresponding Galois group. We describe G-closure data and algebras of finite etale algebras over a general connected ring R in terms of the corresponding finite sets with continuous actions by the etale fundamental group of R. We show that if 2 is invertible, then A n -closure data for free extensions correspond to square roots of the discriminant, and that D4-closure data for quartic monogenic extensions correspond to roots of the cubic resolvent. This is an updated and revised version of the author's PhD thesis.

Étale homotopy types of moduli stacks of polarised abelian schemes

We determine the Artin–Mazur etale homotopy types of moduli stacks of polarised abelian schemes using transcendental methods and derive some arithmetic properties of the etale fundamental groups of these moduli stacks. Finally we analyse the Torelli morphism between the moduli stacks of algebraic curves and principally polarised abelian schemes from an etale homotopy point of view.