Finiteness of algebraic fundamental groups

Fundamental groups of fake projective planes fall into fifty distinct isomorphism classes, one for each complex conjugate pair. We prove that this is not the case for their algebraic fundamental groups: there are only forty-six isomorphism classes. We show that there are four pairs of complex conjugate pairs of fake projective planes that are \operatorname{Aut}(\mathbb{C}) -equivalent and hence have mutually isomorphic algebraic fundamental groups. All other pairs of algebraic fundamental groups are shown to be distinct through explicit finite étale covers. As a by-product, this provides the first examples of commensurable but nonisomorphic lattices in a rank one semisimple Lie group that have isomorphic profinite completions.

The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more recent applications of weights, e.g. to Hasse principles and the computation of motivic cohomology, and discuss some open questions.

In this note, we will give some remarks on the results in [J. Amer. Math. Soc. 24 (2011), no. 3, pp. 709–769] by Hain. In particular, we consider the universal curves M g , n + 1 → M g , n {\mathcal {M}}_{g,n+1}\to {\mathcal {M}}_{g,n} and the sections of their algebraic fundamental groups.

This paper proves the remaining open case of Abhyankar's higher dimensional conjecture on local fundamental groups in characteristic p ((Ab2), (Ab3)). This conjecture, which is analogous to Abhyankar's conjectures on global fundamental groups, proposed that a finite group G is a Galois group over k((x1 ,...,x n))((x1 ··· xr) −1 ) if and only if its maximal prime-to-p quotient is, pro- vided n ≥ 2a nd 1≤ r ≤ n .F or r> 1, this conjecture was disproven in (HP). Here we prove that the conjecture is true in the case r = 1. So the Galois groups over k((x1 ,...,x n))(x −1 1 ) are precisely the cyclic-by-quasi-p groups.

Lie groups and algebraic groups are important in many major areas of mathematics and mathematical physics. We find them in diverse roles, notably as groups of automorphisms of geometric structures, as symmetries of differential systems, or as basic tools in the theory of automorphic forms. The author looks at their development, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. Starting from Lie's theory of local analytic transformation groups and early work on Lie algebras, he follows the process of globalization in its two main frameworks: differential geometry and topology on one hand, algebraic geometry on the other. Chapters II to IV are devoted to the former, Chapters V to VIII, to the latter. The essays in the first part of the book survey various proofs of the full reducibility of linear representations of \\mathbf{SL}_2{(\\mathbb{C})}, the contributions of H. Weyl to representations and invariant theory for semisimple Lie groups, and conclude with a chapter on E. Cartan's theory of symmetric spaces and Lie groups in the large. The second part of the book first outlines various contributions to linear algebraic groups in the 19th century, due mainly to E. Study, E. Picard, and above all, L. Maurer. After being abandoned for nearly fifty years, the theory was revived by C. Chevalley and E. Kolchin, and then further developed by many others. This is the focus of Chapter VI. The book concludes with two chapters on the work of Chevalley on Lie groups and Lie algebras and of Kolchin on algebraic groups and the Galois theory of differential fields, which put their contributions to algebraic groups in a broader context. Professor Borel brings a unique perspective to this study. As an important developer of some of the modern elements of both the differential geometric and the algebraic geometric sides of the theory, he has a particularly deep understanding of the underlying mathematics. His lifelong involvement and his historical research in the subject area give him a special appreciation of the story of its development.

A Lefschetz type theorem is proven relating the algebraic fundamental group of a smooth projective variety X X to the algebraic fundamental group of a subvariety set theoretically defined by ≤ dim ⁡ ( X ) − 2 \le \dim (X)-2 forms.

Let X be a smooth connected projective curve defined over an algebraically closed field k of characteristic p >0. Let G be a finite group whose order is divisible by p. Suppose that G has a normal p-Sylow subgroup. We give a necessary and sufficient condition for G to be a quotient of the algebraic fundamental group pi(1)(X) of X.

Let Mo,, denote the moduli space of Riemann spheres with n ordered marked points. In this article we define the group Out# of quasispecial symmetric outer automorphisms of the algebraic fundamental group 1r1(Mo,n) for all n > 4 to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of Irl(M4,n) and commuting with the group of outer automorphisms of rl (MO,n) obtained by permuting the marked points. Our main result states that Outd is isomorphic to the Grothendieck-Teichmiiller group GT for all n > 5.

Inter-Universal Teichmüller theory applies mono-anabelian algorithms to algebraic fundamental groups in one universe to obtain descriptions of objects constructed from those groups in another. Ivan Fesenko describes Shinichi Mochizuki's theory and new concepts such as frobeniods and Hodge theatres.

Vector bundles on p-adic curves and parallel transport

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than those coming from deformations of "abelian" Galois representations induced by the Tate module of associated Jacobian variety. We develop an arithmetic deformation theory of graded Lie algebras with finite dimensional graded components to serve our purpose.

Covering group theory for topological groups

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> be a field of characteristic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> a proper, smooth, geometrically connected curve over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math>, and 0 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>∞</mml:mi></mml:math> two <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math>-rational points on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math>. We show that any representation of the local Galois group at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>∞</mml:mi></mml:math> extends to a representation of the fundamental group of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>C</mml:mi><mml:mo>-</mml:mo><mml:mo>{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>∞</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:math> which is tamely ramified at 0, provided either that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math> is separately closed or that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi mathvariant="bold">P</mml:mi></mml:mrow> <mml:mn>1</mml:mn> </mml:msup></mml:math>. In the latter case, we show there exists a unique such extension, called “canonical”, with the property that the image of the geometric fundamental group has a unique <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-Sylow subgroup. As an application, we give a global cohomological construcion of the Swan representation in equal characteristic.

Let S be a KLT threefold singularity over an algebraically closed field of positive characteristic p > 5. We prove that its local tale fundamental group is tame and finite.Further, we show that every finite unipotent torsor over a big open subset of S is realized as the restriction of a finite unipotent torsor over S.

Some progress is made in the calculation of local fundamental groups.