On base change of the fundamental group scheme

AbstractLet X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

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On the Bumpy Fundamental Group Scheme

Analytic and Algebraic Geometry ◽

10.1007/978-981-10-5648-2_1 ◽

2017 ◽

pp. 1-17

Author(s):

Marco Antei

Keyword(s):

Fundamental Group ◽

Group Scheme

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Uniformizing the Moduli Stacks of Global G-Shtukas

International Mathematics Research Notices ◽

10.1093/imrn/rnz223 ◽

2019 ◽

Cited By ~ 1

Author(s):

E Arasteh Rad ◽

Urs Hartl

Keyword(s):

Finite Type ◽

Moduli Spaces ◽

Function Field ◽

Group Scheme ◽

Base Change ◽

Generic Fiber ◽

Smooth Projective Curve ◽

Langlands Program ◽

Moduli Stacks ◽

Divisible Groups

Abstract This is the 2nd in a sequence of articles, in which we explore moduli stacks of global $\mathfrak{G}$-shtukas, the function field analogs for Shimura varieties. Here $\mathfrak{G}$ is a flat affine group scheme of finite type over a smooth projective curve $C$ over a finite field. Global $\mathfrak{G}$-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global $\mathfrak{G}$-shtukas are algebraic Deligne–Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the 1st article we explained the relation between global $\mathfrak{G}$-shtukas and local ${{\mathbb{P}}}$-shtukas, which are the function field analogs of $p$-divisible groups. Here ${{\mathbb{P}}}$ is the base change of $\mathfrak{G}$ to the complete local ring at a point of $C$. When ${{\mathbb{P}}}$ is smooth with connected reductive generic fiber we proved the existence of Rapoport–Zink spaces for local ${{\mathbb{P}}}$-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global $\mathfrak{G}$-shtukas for smooth $\mathfrak{G}$ with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands–Rapoport conjecture for our moduli stacks.

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MODELS OF TORSORS AND THE FUNDAMENTAL GROUP SCHEME

Nagoya Mathematical Journal ◽

10.1017/nmj.2016.67 ◽

2016 ◽

Vol 230 ◽

pp. 18-34 ◽

Cited By ~ 1

Author(s):

MARCO ANTEI ◽

MICHEL EMSALEM

Keyword(s):

Finite Number ◽

Fundamental Group ◽

Galois Group ◽

Closed Subset ◽

Valuation Ring ◽

Group Scheme ◽

Discrete Valuation Ring ◽

Generic Fiber ◽

Natural Morphism ◽

Tannakian Category

Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X\rightarrow S$, this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_{\unicode[STIX]{x1D702}}$ to the generic fiber of the fundamental group scheme of $X$. Given a torsor $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under an affine group scheme $G$ over the generic fiber of $X$, we address the question of finding a model of this torsor over $X$, focusing in particular on the case where $G$ is finite. We provide several answers to this question, showing for instance that, when $X$ is integral and regular of relative dimension 1, such a model exists on some model $X^{\prime }$ of $X_{\unicode[STIX]{x1D702}}$ obtained by performing a finite number of Néron blowups along a closed subset of the special fiber of $X$. Furthermore, we show that when $G$ is étale, then we can find a model of $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under the action of some smooth group scheme. In the first part of the paper, we show that the relative fundamental group scheme of $X$ has an interpretation as the Tannaka Galois group of a Tannakian category constructed starting from the universal torsor.

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