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Richard Pink

Bio: Richard Pink is an academic researcher from ETH Zurich. The author has contributed to research in topics: Galois module & Galois cohomology. The author has an hindex of 23, co-authored 62 publications receiving 1952 citations. Previous affiliations of Richard Pink include University of Mannheim & University of Bonn.


Papers
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Journal ArticleDOI
TL;DR: In this article, it was shown that every finite subgroup of a field over a field of any characteristic has a subgroup which is composed of finite simple groups of Lie type in characteristic p, a commutative group of order prime to p, and a p-group.
Abstract: Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of GLn over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic p, a commutative group of order prime to p, and a p-group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.

178 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the algebraic monodromy groups associated to a motive over a number field are generated by certain one-parameter subgroups determined by Hodge numbers.
Abstract: We prove that the `-adic algebraic monodromy groups associated to a motive over a number field are generated by certain one-parameter subgroups determined by Hodge numbers. In the special case of an abelian variety we obtain stronger statements saying roughly that the `-adic algebraic monodromy groups look like a Mumford-Tate group of some (other?) abelian variety. When the endomorphism ring is Z and the dimension satisfies certain numerical conditions, we deduce the Mumford-Tate conjecture for this abelian variety. We also discuss the problem of finding places of ordinary reduction.

127 citations

Journal ArticleDOI
TL;DR: In this article, the authors give a qualitative classification of all compact subgroups Γ ⊂ GL n (F ), where F is a local field and n is arbitrary, up to finite index and a finite number of abelian subquotients.

114 citations

Book ChapterDOI
01 Jan 2005
TL;DR: In this paper, a conjecture combining the Mordell-Lang conjecture with an important special case of the Andre-Oort conjecture is proposed, and existing results imply evidence for it.
Abstract: We propose a conjecture combining the Mordell-Lang conjecture with an important special case of the Andre-Oort conjecture, and explain how existing results imply evidence for it.

105 citations


Cited by
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Journal ArticleDOI
TL;DR: In this article, the authors prove an automorphy lifting theorem for l-adic representations where they impose a new condition at l, which they call "potentential diagonalizability".
Abstract: We prove an automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call \potentential diagonalizability." This result allows for \change of weight" and seems to be substantially more exible than previous theorems along the same lines. We derive several applications. For instance we show that any irreducible, totally odd, essentially self-dual, regular, weakly compatible system of l-adic representations of the absolute Galois group of a totally real eld is potentially automorphic and hence is pure and its L-function has meromorphic continuation to the whole complex plane and satises the expected functional equation.

315 citations

Book
21 Sep 2017
TL;DR: The first comprehensive introduction to the theory of algebraic group schemes over fields was given in this paper, which includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry.
Abstract: Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

233 citations