Algebraic Tori and Essential Dimension
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
The Book of Involutions
The Book of Involutions Max-Albert Knus Alexander Sergejvich Merkurjev Herbert Markus Rost Jean-Pierre Tignol @ @ @ @ @ @ @ @ The Book of Involutions Max-Albert Knus Alexander Merkurjev Markus Rost Jean-Pierre Tignol Author address: Dept. Mathematik, ETH-Zentrum, CH-8092 Zurich,¨ Switzerland E-mail address: [email protected] URL: http://www.math.ethz.ch/~knus/ Dept. of Mathematics, University of California at Los Angeles, Los Angeles, California, 90095-1555, USA E-mail address: [email protected] URL: http://www.math.ucla.edu/~merkurev/ NWF I - Mathematik, Universitat¨ Regensburg, D-93040 Regens- burg, Germany E-mail address: [email protected] URL: http://www.physik.uni-regensburg.de/~rom03516/ Departement´ de mathematique,´ Universite´ catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium E-mail address: [email protected] URL: http://www.math.ucl.ac.be/tignol/ Contents Pr´eface . ix Introduction . xi Conventions and Notations . xv Chapter I. Involutions and Hermitian Forms . 1 1. Central Simple Algebras . 3 x 1.A. Fundamental theorems . 3 1.B. One-sided ideals in central simple algebras . 5 1.C. Severi-Brauer varieties . 9 2. Involutions . 13 x 2.A. Involutions of the first kind . 13 2.B. Involutions of the second kind . 20 2.C. Examples . 23 2.D. Lie and Jordan structures . 27 3. Existence of Involutions . 31 x 3.A. Existence of involutions of the first kind . 32 3.B. Existence of involutions of the second kind . 36 4. Hermitian Forms . 41 x 4.A. Adjoint involutions . 42 4.B. Extension of involutions and transfer . -
Essential Dimension of Finite Groups
University of California Los Angeles Essential Dimension of Finite Groups A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Wanshun Wong 2012 c Copyright by Wanshun Wong 2012 Abstract of the Dissertation Essential Dimension of Finite Groups by Wanshun Wong Doctor of Philosophy in Mathematics University of California, Los Angeles, 2012 Professor Alexander Merkurjev, Chair In this thesis we study the essential dimension of the first Galois cohomology functors of finite groups. Following the result by N. A. Karpenko and A. S. Merkurjev about the essential dimension of finite p-groups over a field containing a primitive p-th root of unity, we compute the essential dimension of finite cyclic groups and finite abelian groups over a field containing all primitive p-th roots of unity for all prime divisors p of the order of the group. For the computation of the upper bound of the essential dimension we apply the techniques about affine group schemes, and for the lower bound we make use of the tools of canonical dimension and fibered categories. We also compute the essential dimension of small groups over the field of rational numbers to illustrate the behavior of essential dimension when the base field does not contain all relevant primitive roots of unity. ii The dissertation of Wanshun Wong is approved. Amit Sahai Richard Elman Paul Balmer Alexander Merkurjev, Committee Chair University of California, Los Angeles 2012 iii Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: -
Arxiv:1906.01274V1 [Math.NT] 4 Jun 2019 Rpsto .] Aqe Ishmefas Rvdsoepofi I Yale 1.4.1]
CHOW’S THEOREM FOR SEMI-ABELIAN VARIETIES AND BOUNDS FOR SPLITTING FIELDS OF ALGEBRAIC TORI CHIA-FU YU Abstract. A theorem of Chow concerns homomorphisms of two abelian vari- eties under a primary field extension base change. In this paper we generalize Chow’s theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori. 1. Introduction Let k be a field, k¯ an algebraic closure of k, and ks the separable closure of k in k¯. A connected algebraic k-group T is an algebraic torus if there is a k¯-isomorphism d T ⊗k k¯ ≃ (Gm) ⊗k k of algebraic groups for some integer d ≥ 0. We say T splits d over a field extension K of k if there is a K-isomorphism T ⊗k K ≃ (Gm) ⊗k K. This paper is motivated from the following fundamental result. Theorem 1.1. Any algebraic k-torus T splits over ks. In other words, T splits over a finite separable field extension of k. This theorem is well known and it is stated and proved in the literature several times. Surprisingly, different authors choose their favorite proofs which are all quite different. As far as we know, the first proof is given by Takashi Ono [40, Proposition 1.2.1]. Armand Borel gives a different proof in his book Linear Algebraic Groups; see [3, Proposition 8.11]. In the second edition of his book Linear Algebraic Groups [49], T.A. -
Material on Algebraic and Lie Groups
2 Lie groups and algebraic groups. 2.1 Basic Definitions. In this subsection we will introduce the class of groups to be studied. We first recall that a Lie group is a group that is also a differentiable manifold 1 and multiplication (x, y xy) and inverse (x x ) are C1 maps. An algebraic group is a group7! that is also an algebraic7! variety such that multi- plication and inverse are morphisms. Before we can introduce our main characters we first consider GL(n, C) as an affi ne algebraic group. Here Mn(C) denotes the space of n n matrices and GL(n, C) = g Mn(C) det(g) =) . Now Mn(C) is given the structure nf2 2 j 6 g of affi ne space C with the coordinates xij for X = [xij] . This implies that GL(n, C) is Z-open and as a variety is isomorphic with the affi ne variety 1 Mn(C) det . This implies that (GL(n, C)) = C[xij, det ]. f g O Lemma 1 If G is an algebraic group over an algebraically closed field, F , then every point in G is smooth. Proof. Let Lg : G G be given by Lgx = gx. Then Lg is an isomorphism ! 1 1 of G as an algebraic variety (Lg = Lg ). Since isomorphisms preserve the set of smooth points we see that if x G is smooth so is every element of Gx = G. 2 Proposition 2 If G is an algebraic group over an algebraically closed field F then the Z-connected components Proof. -
Algebraic Tori As Degenerations of Abelian Varieties
Mathematika 64 (2018) 303–329 c 2018 University College London doi:10.1112/S0025579317000420 ALGEBRAIC TORI AS DEGENERATIONS OF ABELIAN VARIETIES KAI-WEN LAN AND JUNECUE SUH Abstract. We first show that every algebraic torus over any field, not necessarily split, can be realized as the special fiber of a semi-abelian scheme whose generic fiber is an absolutely simple abelian variety. Then we investigate which algebraic tori can be thus obtained, when we require the generic fiber of the semi-abelian scheme to carry non-trivial endomorphism structures. Contents 1 Introduction 303 2 One-dimensional case 305 3 Descent data for tori 307 4 Theory of degeneration 308 5 Conditions for degenerations 310 6 Proof of main theorem 316 7 Non-trivial endomorphisms 319 Acknowledgements 329 References 329 §1. Introduction. We start with a well-known example, the Dwork family of elliptic curves X 3 C Y 3 C Z 3 − 3t XY Z D 0 1 t over the projective line PQ over Q, where is the affine coordinate, with the identity section given by the point at infinity TX V Y V ZUDT1 V −1 V 0U. It also doubles as the modular curve of level 3. It has bad, semistable reduction at t D 1 and the third roots of unity. The corresponding Neron´ model has the split torus as the identity component of the fiber at t D 1 (because the equation becomes XYZ D 0), but at t D 1 we get a non-split torus T that sits in the short exact sequence / / p Norm / / ; 1 T ResQ. -
Algebraic Tori: a Computational Approach
Western University Scholarship@Western Electronic Thesis and Dissertation Repository 8-8-2017 12:00 AM Algebraic Tori: A Computational Approach Armin Jamshidpey The University of Western Ontario Supervisor Prof. Nicole Lemire The University of Western Ontario Joint Supervisor Prof. Eric Schost The University of Western Ontario Graduate Program in Mathematics A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Armin Jamshidpey 2017 Follow this and additional works at: https://ir.lib.uwo.ca/etd Recommended Citation Jamshidpey, Armin, "Algebraic Tori: A Computational Approach" (2017). Electronic Thesis and Dissertation Repository. 4692. https://ir.lib.uwo.ca/etd/4692 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract The rationality problem for algebraic tori is well known. It is known that any algebraic torus is unirational over its field of definition. The first purpose of this work is to solve rational- ity problem for 5 dimensional stably rational algebraic tori with an indecomposable character lattice. In order to do so, we have studied the associated character lattices of the mentioned algebraic tori. For each character lattice L, either we see the lattice as an associated lattice to a root system (of which rationality of its corresponding algebraic torus is known) or we find a reduced component of L so that we can relate rationality of the associated algebraic torus to lower dimensions. -
On the Essential Dimension of Cyclic P-Groups
1 On the essential dimension of cyclic p-groups Mathieu Florence1 December 2006 Abstract. Let p be a prime number and r ≥ 1 an integer. We compute the essential dimension of Z/prZ over fields of characteristic not p, containing the p-th roots of unity (theorem 3.1). In particular, we have edQ(Z/8Z) = 4, a result which was conjectured by Buhler and Reichstein in 1995 (unpublished). Keywords and Phrases: Galois cohomology, essential dimension, Severi-Brauer varieties. A la m´emoire de mon p`ere. Acknowledgements We thank Jean Fasel and Giordano Favi for fruitful discussions about essential dimension. Contents Acknowledgements 1 1. Introduction 1 2. Some auxiliary results 2 3. The main theorem 5 References 6 1. Introduction The notion of essential dimension was introduced by Buhler and Reichstein for finite groups in [BR]. It was later generalized by Reichstein to arbitrary linear algebraic groups ([Re]). Throughout this paper, we shall assume that the reader is somewhat familiar with this concept. A convenient and comprehensive reference on this subject is [BF]. Definition 1.1. Let k be a field, and G a (smooth) linear algebraic k-group. The essential dimension of G over k, denoted by edk(G), is the smallest nonnegative integer n with the following property. For each field extension K/k, and each GK -torsor T −→ Spec(K), there exists a ′ ′ ′ subfield K of K, containing k, and a GK′ -torsor T −→ Spec(K ), such that ′ i) The GK -torsors T and TK are isomorphic, ii) The transcendence degree of K′/k is equal to n. -
Underlying Varieties and Group Structures 3
UNDERLYING VARIETIES AND GROUP STRUCTURES VLADIMIR L. POPOV Abstract. Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure. 1. Introduction. This paper arose from a short preprint [16] and is its extensive extension. As starting point of [14] served the question of B. Kunyavsky [10] about the validity of the statement, which is for- mulated below as Corollary of Theorem 1. This statement concerns the possibility to present the underlying variety of a connected reductive algebraic group in the form of a product of some special algebraic va- rieties. Sections 2, 3 make up the content of [16], where the possibility of such presentations is explored. For some of them, in Theorem 1 is proved their existence, and in Theorems 2–5, on the contrary, their non-existence. Theorem 1 shows that there are non-isomorphic reductive groups whose underlying varieties are isomorphic. In Sections 4–10, we explore the problem, naturally arising in connection with this, as to what ex- tent the underlying variety of an algebraic group determines its group structure. In Theorems 6–8, it is shown that some group properties (dimension of unipotent radical, reductivity, solvability, unipotency, arXiv:2105.12861v1 [math.AG] 26 May 2021 toroidality in the sense of Rosenlicht) are equivalent to certain geomet- ric properties of the underlying group variety. Theorem 8 generalizes to solvable groups M. -
1:34 Pm April 11, 2013 Red.Tex
1:34 p.m. April 11, 2013 Red.tex The structure of reductive groups Bill Casselman University of British Columbia, Vancouver [email protected] An algebraic group defined over F is an algebraic variety G with group operations specified in algebraic terms. For example, the group GLn is the subvariety of (n + 1) × (n + 1) matrices A 0 0 a with determinant det(A) a = 1. The matrix entries are well behaved functions on the group, here for 1 example a = det− (A). The formulas for matrix multiplication are certainly algebraic, and the inverse of a matrix A is its transpose adjoint times the inverse of its determinant, which are both algebraic. Formally, this means that we are given (a) an F •rational multiplication map G × G −→ G; (b) an F •rational inverse map G −→ G; (c) an identity element—i.e. an F •rational point of G. I’ll look only at affine algebraic groups (as opposed, say, to elliptic curves, which are projective varieties). In this case, the variety G is completely characterized by its affine ring AF [G], and the data above are respectively equivalent to the specification of (a’) an F •homomorphism AF [G] −→ AF [G] ⊗F AF [G]; (b’) an F •involution AF [G] −→ AF [G]; (c’) a distinguished homomorphism AF [G] −→ F . The first map expresses a coordinate in the product in terms of the coordinates of its terms. For example, in the case of GLn it takes xik −→ xij ⊗ xjk . j In addition, these data are subject to the group axioms. I’ll not say anything about the general theory of such groups, but I should say that in practice the specification of an algebraic group is often indirect—as a subgroup or quotient, say, of another simpler one. -
Essential Dimension
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 4, October 2017, Pages 635–661 http://dx.doi.org/10.1090/bull/1564 Article electronically published on December 19, 2016 ESSENTIAL DIMENSION ALEXANDER S. MERKURJEV Abstract. In this paper we survey research on the essential dimension that was introduced by J. Buhler and Z. Reichstein. Informally speaking, the es- sential dimension of a class of algebraic objects is the minimal number of algebraically independent parameters one needs to define any object in the class. The notion of essential dimension, which is defined in elementary terms, has surprising connections to many areas of algebra, such as algebraic geome- try, algebraic K-theory, Galois cohomology, representation theory of algebraic groups, theory of fibered categories and valuation theory. The highlights of the survey are the computations of the essential dimensions of finite groups, groups of multiplicative type and the spinor groups. 1. Introduction The essential dimension of an algebraic object is an integer that measures the complexity of the object. For example, let Q =(a, b)K be a quaternion algebra over a field extension K of a base field F of characteristic not 2. That is, let Q be a four-dimensional algebra over K with basis {1,i,j,ij} and multiplication table i2 = a, j2 = b, ij = −ji,wherea and b are nonzero elements in K.Thus,Q is determined by two parameters a and b. Equivalently, we can say that the algebra Q is defined over the subfield K = F (a, b)ofK,namely,Q Q ⊗K K for the quaternion algebra Q =(a, b)K over the field K of transcendence degree at most 2overF . -
Essential Dimension, Spinor Groups and Quadratic Forms 3
ESSENTIAL DIMENSION, SPINOR GROUPS, AND QUADRATIC FORMS PATRICK BROSNANy, ZINOVY REICHSTEINy, AND ANGELO VISTOLIz Abstract. We prove that the essential dimension of the spinor group Spinn grows exponentially with n and use this result to show that qua- dratic forms with trivial discriminant and Hasse-Witt invariant are more complex, in high dimensions, than previously expected. 1. Introduction Let K be a field of characteristic different from 2 containing a square root of −1, W(K) be the Witt ring of K and I(K) be the ideal of classes of even-dimensional forms in W(K); cf. [Lam73]. By abuse of notation, we will write q 2 Ia(K) if the Witt class on the non-degenerate quadratic form q defined over K lies in Ia(K). It is well known that every q 2 Ia(K) can be expressed as a sum of the Witt classes of a-fold Pfister forms defined over K; see, e.g., [Lam73, Proposition II.1.2]. If dim(q) = n, it is natural to ask how many Pfister forms are needed. When a = 1 or 2, it is easy to see that n Pfister forms always suffice; see Proposition 4.1. In this paper we will prove the following result, which shows that the situation is quite different when a = 3. Theorem 1.1. Let k be a field of characteristic different from 2 and n ≥ 2 be an even integer. Then there is a field extension K=k and an n-dimensional quadratic form q 2 I3(K) with the following property: for any finite field extension L=K of odd degree qL is not Witt equivalent to the sum of fewer than 2(n+4)=4 − n − 2 7 3-fold Pfister forms over L. -
GVF NOTES/ ITAฯ BEN YAACOV, E.H. Contents 1. Continuous Logic 2 1.1
GVF NOTES/ ITAÏ BEN YAACOV, E.H. Contents 1. Continuous logic 2 1.1. Ultraproducts 2 1.2. Formulas 2 2. Measure theory 3 2.7. Positive linear functionals 5 3. Valued fields 6 3.1. Generically stable types 6 3.2. Comparison of V“ with V an 6 3.4. The linear part of a generically stable type 6 3.6. valuative volumes and mixed volumes 7 3.9. Minkowski’s mixed volumes 9 4. Multiply valued fields 10 4.6. Topology on V ALF 12 4.7. Admissible measures 12 4.11. Induced structure on a subfield 13 4.12. Renormalization 13 4.19. Proper subvarieties 15 4.21. Extending valuations on subrings 15 4.26. Symmetric extensions 17 4.29. Heights 18 4.34. A criterion for globalization 18 4.35. GVF’s as continuous logic structures 18 4.38. Archimedean valuations 19 5. Classical GVF structures. 21 6. Conjectures 22 a 7. Gm-fullness and the GVF structure on K . 23 7.1. Axiomatizability of Gm-fullness. 24 8. Adelic canonical amalgamation 25 9. One variable 32 10. 1-types of height zero 33 10.1. Local case 34 10.2. The purely non-archimedean case 34 References 39 Date: January 26, 2016. 1 2 GVF NOTES/ ITAÏ BEN YAACOV, E.H. 1. Continuous logic A language consists of function and relation symbols as usual. A structure is a set (or a family of sets in the many-sorted situation) with function symbols and terms treated as usual. But the interpretation of a relation symbol R(x1; : : : ; xn) n is a function A ! R.