Related papers: Anabelian geometry with etale homotopy types

We discuss how canonical and universal constructions, properties and characterizations interact with equality in the framework of Homotopy Type Theory, comparing it with Grothendieck's use of equality and shedding further light on…

The pro-\'etale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes formerly known fundamental groups -- the usual \'etale fundamental group $\pi_1^{\mathrm{et}}$ defined in SGA1 and the more general group defined in…

Those are the notes for the 2015 Summer Research Institute on Algebraic Geometry. We report on Deligne's finiteness theorem for $\ell$-adic representations on smooth varieties defined over a finite field, on its crystalline version, and on…

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…

We study the homeomorphism types of certain covers of (always orientable) surfaces, usually of infinite-type. We show that every surface with non-abelian fundamental group is covered by every noncompact surface, we identify the universal…

$\nabla$-algebra is a natural generalization of Heyting algebra, unifying many algebraic structures including bounded lattices, Heyting algebras, temporal Heyting algebras and the algebraic presentation of the dynamic topological systems.…

Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Its dual group $\hat{G}$ is a split reductive group over $\mathbb{Z}$.…

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…

We show the vanishing of the second homotopy group of the \'etale homotopy type of a smooth connected algebraic group over a separably closed field, completed away from the characteristic. This is an algebraic analogue of a classical…

Let A be an abelian surface over F_q, the field of q elements. The rational points on A/\F_q form an abelian group A(\F_q) \simeq \Z/n_1\Z \times \Z/n_1 n_2 \Z \times \Z/n_1 n_2 n_3\Z \times\Z/n_1 n_2 n_3 n_4\Z. We are interested in knowing…

A natural topology on the space of left orderings of an arbitrary semi-group is introduced. It is proved that this space is compact and that for free abelian groups it is homeomorphic to the Cantor set. An application of this result is a…

We show that the existence of rational points on smooth varieties over a field can be detected using homotopy fixed points of etale topological types under the Galois action. As our main example we show that the surjectivity statement in…

We construct new examples of singular projective plane curves whose complements have finite and non-abelian fundamental groups, by generalizing the classical three cuspidal quartic curve discovered by Zariski.

Given a Lie algebroid we discuss the existence of a smooth abelian integration of its abelianization. We show that the obstructions are related to the extended monodromy groups introduced recently in \cite{CFMb}. We also show that this…

We show that the continuous \'etale cohomology groups $H^n_{\mathrm{cont}}(X,\mathbf{Z}_l(n))$ of smooth varieties $X$ over a finite field $k$ are spanned as $\mathbf{Z}_l$-modules by the $n$-th Milnor $K$-sheaf locally for the Zariski…

Generalizing homogeneous spectra for rings graded by natural numbers, we introduce multihomogeneous spectra for rings graded by abelian groups. Such homogeneous spectra have the same completeness properties as their classical counterparts,…

Let $K$ be the function field of a smooth projective geometrically integral curve over a finite extension of $\mathbb{Q}_p$. Following the works of Harari, Scheiderer, Szamuely, Izquierdo, and Tian, we study the local-global and weak…

In this paper, we present some new results on the geometrically m-step solvable Grothendieck conjecture in anabelian geometry. Specifically, we show the (weak bi-anabelian and strong bi-anabelian) geometrically m-step solvable Grothendieck…

The affine line and the punctured affine line over a finite field F are taken as benchmarks for the problem of describing geometric \'etale fundamental groups. To this end, using a reformulation of Tannaka duality we construct for a…

I show that one can explicitly construct topologically/geometrically distinguishable data which provide isomorphic copies (i.e. \emph{isomorphs}) of the tempered fundamental group of a geometrically connected, smooth, quasi-projective…