Some Aspects of the Algebraic Theory of Quadratic Forms
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MASTER COURSE: Quaternion Algebras and Quadratic Forms Towards Shimura Curves
MASTER COURSE: Quaternion algebras and Quadratic forms towards Shimura curves Prof. Montserrat Alsina Universitat Polit`ecnicade Catalunya - BarcelonaTech EPSEM Manresa September 2013 ii Contents 1 Introduction to quaternion algebras 1 1.1 Basics on quaternion algebras . 1 1.2 Main known results . 4 1.3 Reduced trace and norm . 6 1.4 Small ramified algebras... 10 1.5 Quaternion orders . 13 1.6 Special basis for orders in quaternion algebras . 16 1.7 More on Eichler orders . 18 1.8 Eichler orders in non-ramified and small ramified Q-algebras . 21 2 Introduction to Fuchsian groups 23 2.1 Linear fractional transformations . 23 2.2 Classification of homographies . 24 2.3 The non ramified case . 28 2.4 Groups of quaternion transformations . 29 3 Introduction to Shimura curves 31 3.1 Quaternion fuchsian groups . 31 3.2 The Shimura curves X(D; N) ......................... 33 4 Hyperbolic fundamental domains . 37 4.1 Groups of quaternion transformations and the Shimura curves X(D; N) . 37 4.2 Transformations, embeddings and forms . 40 4.2.1 Elliptic points of X(D; N)....................... 43 4.3 Local conditions at infinity . 48 iii iv CONTENTS 4.3.1 Principal homotheties of Γ(D; N) for D > 1 . 48 4.3.2 Construction of a fundamental domain at infinity . 49 4.4 Principal symmetries of Γ(D; N)........................ 52 4.5 Construction of fundamental domains (D > 1) . 54 4.5.1 General comments . 54 4.5.2 Fundamental domain for X(6; 1) . 55 4.5.3 Fundamental domain for X(10; 1) . 57 4.5.4 Fundamental domain for X(15; 1) . -
On the Linear Transformations of a Quadratic Form Into Itself*
ON THE LINEAR TRANSFORMATIONS OF A QUADRATIC FORM INTO ITSELF* BY PERCEY F. SMITH The problem of the determination f of all linear transformations possessing an invariant quadratic form, is well known to be classic. It enjoyed the atten- tion of Euler, Cayley and Hermite, and reached a certain stage of com- pleteness in the memoirs of Frobenius,| Voss,§ Lindemann|| and LoEWY.^f The investigations of Cayley and Hermite were confined to the general trans- formation, Erobenius then determined all proper transformations, and finally the problem was completely solved by Lindemann and Loewy, and simplified by Voss. The present paper attacks the problem from an altogether different point, the fundamental idea being that of building up any such transformation from simple elements. The primary transformation is taken to be central reflection in the quadratic locus defined by setting the given form equal to zero. This transformation is otherwise called in three dimensions, point-plane reflection,— point and plane being pole and polar plane with respect to the fundamental quadric. In this way, every linear transformation of the desired form is found to be a product of central reflections. The maximum number necessary for the most general case is the number of variables. Voss, in the first memoir cited, proved this theorem for the general transformation, assuming the latter given by the equations of Cayley. In the present paper, however, the theorem is derived synthetically, and from this the analytic form of the equations of trans- formation is deduced. * Presented to the Society December 29, 1903. Received for publication, July 2, 1P04. -
Generically Split Octonion Algebras and A1-Homotopy Theory
Generically split octonion algebras and A1-homotopy theory Aravind Asok∗ Marc Hoyoisy Matthias Wendtz Abstract 1 We study generically split octonion algebras over schemes using techniques of A -homotopy theory. By combining affine representability results with techniques of obstruction theory, we establish classifica- tion results over smooth affine schemes of small dimension. In particular, for smooth affine schemes over algebraically closed fields, we show that generically split octonion algebras may be classified by charac- teristic classes including the second Chern class and another “mod 3” invariant. We review Zorn’s “vector matrix” construction of octonion algebras, generalized to rings by various authors, and show that generi- cally split octonion algebras are always obtained from this construction over smooth affine schemes of low dimension. Finally, generalizing P. Gille’s analysis of octonion algebras with trivial norm form, we observe that generically split octonion algebras with trivial associated spinor bundle are automatically split in low dimensions. Contents 1 Introduction 1 2 Octonion algebras, algebra and geometry 6 2.1 Octonion algebras and Zorn’s vector matrices......................................7 2.2 Octonion algebras and G2-torsors............................................9 2.3 Homogeneous spaces related to G2 and octonions.................................... 11 1 3 A -homotopy sheaves of BNis G2 17 1 3.1 Some A -fiber sequences................................................. 17 1 3.2 Characteristic maps and strictly A -invariant sheaves.................................. 19 3.3 Realization and characteristic maps........................................... 21 1 3.4 Some low degree A -homotopy sheaves of BNis Spinn and BNis G2 .......................... 24 4 Classifying octonion algebras 29 4.1 Classifying generically split octonion algebras...................................... 30 4.2 When are octonion algebras isomorphic to Zorn algebras?............................... -
Quadratic Forms, Lattices, and Ideal Classes
Quadratic forms, lattices, and ideal classes Katherine E. Stange March 1, 2021 1 Introduction These notes are meant to be a self-contained, modern, simple and concise treat- ment of the very classical correspondence between quadratic forms and ideal classes. In my personal mental landscape, this correspondence is most naturally mediated by the study of complex lattices. I think taking this perspective breaks the equivalence between forms and ideal classes into discrete steps each of which is satisfyingly inevitable. These notes follow no particular treatment from the literature. But it may perhaps be more accurate to say that they follow all of them, because I am repeating a story so well-worn as to be pervasive in modern number theory, and nowdays absorbed osmotically. These notes require a familiarity with the basic number theory of quadratic fields, including the ring of integers, ideal class group, and discriminant. I leave out some details that can easily be verified by the reader. A much fuller treatment can be found in Cox's book Primes of the form x2 + ny2. 2 Moduli of lattices We introduce the upper half plane and show that, under the quotient by a natural SL(2; Z) action, it can be interpreted as the moduli space of complex lattices. The upper half plane is defined as the `upper' half of the complex plane, namely h = fx + iy : y > 0g ⊆ C: If τ 2 h, we interpret it as a complex lattice Λτ := Z+τZ ⊆ C. Two complex lattices Λ and Λ0 are said to be homothetic if one is obtained from the other by scaling by a complex number (geometrically, rotation and dilation). -
Classification of Quadratic Surfaces
Classification of Quadratic Surfaces Pauline Rüegg-Reymond June 14, 2012 Part I Classification of Quadratic Surfaces 1 Context We are studying the surface formed by unshearable inextensible helices at equilibrium with a given reference state. A helix on this surface is given by its strains u ∈ R3. The strains of the reference state helix are denoted by ˆu. The strain-energy density of a helix given by u is the quadratic function 1 W (u − ˆu) = (u − ˆu) · K (u − ˆu) (1) 2 where K ∈ R3×3 is assumed to be of the form K1 0 K13 K = 0 K2 K23 K13 K23 K3 with K1 6 K2. A helical rod also has stresses m ∈ R3 related to strains u through balance laws, which are equivalent to m = µ1u + µ2e3 (2) for some scalars µ1 and µ2 and e3 = (0, 0, 1), and constitutive relation m = K (u − ˆu) . (3) Every helix u at equilibrium, with reference state uˆ, is such that there is some scalars µ1, µ2 with µ1u + µ2e3 = K (u − ˆu) µ1u1 = K1 (u1 − uˆ1) + K13 (u3 − uˆ3) (4) ⇔ µ1u2 = K2 (u2 − uˆ2) + K23 (u3 − uˆ3) µ1u3 + µ2 = K13 (u1 − uˆ1) + K23 (u1 − uˆ1) + K3 (u3 − uˆ3) Assuming u1 and u2 are not zero at the same time, we can rewrite this surface (K2 − K1) u1u2 + K23u1u3 − K13u2u3 − (K2uˆ2 + K23uˆ3) u1 + (K1uˆ1 + K13uˆ3) u2 = 0. (5) 1 Since this is a quadratic surface, we will study further their properties. But before going to general cases, let us observe that the u3 axis is included in (5) for any values of ˆu and K components. -
Rolle's Theorem Over Local Fields
Rolle's Theorem over Local Fields Cristina M. Ballantine Thomas R. Shemanske April 11, 2002 Abstract In this paper we show that no non-archimedean local field has Rolle's property. 1 Introduction Rolle's property for a field K is that if f is a polynomial in K[x] which splits over K, then its derivative splits over K. This property is implied by the usual Rolle's theorem taught in calculus for functions over the real numbers, however for fields with no ordering, it is the best one can hope for. Of course, Rolle's property holds not only for the real numbers, but also for any algebraically- or real-closed field. Kaplansky ([3], p. 30) asks for a characterization of all such fields. For finite fields, Rolle's property holds only for the fields with 2 and 4 elements [2], [1]. In this paper, we show that Rolle's property fails to hold over any non-archimedean local field (with a finite residue class field). In particular, such fields include the completion of any global field of number theory with respect to a nontrivial non-archimedean valuation. Moreover, we show that there are counterexamples for Rolle's property for polynomials of lowest possible degree, namely cubics. 2 Rolle's Theorem Theorem 2.1. Rolle's property fails to hold over any non-archimedean local field having finite residue class field. Proof. Let K be a non-archimedean local field with finite residue class field. Let O be the ring of integers of K, and P its maximal ideal. -
Local-Global Methods in Algebraic Number Theory
LOCAL-GLOBAL METHODS IN ALGEBRAIC NUMBER THEORY ZACHARY KIRSCHE Abstract. This paper seeks to develop the key ideas behind some local-global methods in algebraic number theory. To this end, we first develop the theory of local fields associated to an algebraic number field. We then describe the Hilbert reciprocity law and show how it can be used to develop a proof of the classical Hasse-Minkowski theorem about quadratic forms over algebraic number fields. We also discuss the ramification theory of places and develop the theory of quaternion algebras to show how local-global methods can also be applied in this case. Contents 1. Local fields 1 1.1. Absolute values and completions 2 1.2. Classifying absolute values 3 1.3. Global fields 4 2. The p-adic numbers 5 2.1. The Chevalley-Warning theorem 5 2.2. The p-adic integers 6 2.3. Hensel's lemma 7 3. The Hasse-Minkowski theorem 8 3.1. The Hilbert symbol 8 3.2. The Hasse-Minkowski theorem 9 3.3. Applications and further results 9 4. Other local-global principles 10 4.1. The ramification theory of places 10 4.2. Quaternion algebras 12 Acknowledgments 13 References 13 1. Local fields In this section, we will develop the theory of local fields. We will first introduce local fields in the special case of algebraic number fields. This special case will be the main focus of the remainder of the paper, though at the end of this section we will include some remarks about more general global fields and connections to algebraic geometry. -
ABSTRACT THEORY of SEMIORDERINGS 1. Introduction
ABSTRACT THEORY OF SEMIORDERINGS THOMAS C. CRAVEN AND TARA L. SMITH Abstract. Marshall’s abstract theory of spaces of orderings is a powerful tool in the algebraic theory of quadratic forms. We develop an abstract theory for semiorderings, developing a notion of a space of semiorderings which is a prespace of orderings. It is shown how to construct all finitely generated spaces of semiorder- ings. The morphisms between such spaces are studied, generalizing the extension of valuations for fields into this context. An important invariant for studying Witt rings is the covering number of a preordering. Covering numbers are defined for abstract preorderings and related to other invariants of the Witt ring. 1. Introduction Some of the earliest work with formally real fields in quadratic form theory, follow- ing Pfister’s revitalization of this area of research, involved Witt rings of equivalence classes of nondegenerate quadratic forms and the spaces of orderings which are closely associated with the prime ideals of the Witt rings. A ring theoretic approach was pioneered by Knebusch, Rosenberg and Ware in [20]. The spaces of orderings were studied in [6] and all Boolean spaces were shown to occur. A closer tie to the theory of quadratic forms was achieved by M. Marshall who developed an abstract theory of spaces of orderings to help in studying the reduced Witt rings of formally real fields (cf. [22]). Marshall’s theory was more closely related to what actually happens with Witt rings of fields than the work in [20] (and this was later put into a ring-theoretic context by Rosenberg and Kleinstein [18]). -
Gauging the Octonion Algebra
UM-P-92/60_» Gauging the octonion algebra A.K. Waldron and G.C. Joshi Research Centre for High Energy Physics, University of Melbourne, Parkville, Victoria 8052, Australia By considering representation theory for non-associative algebras we construct the fundamental and adjoint representations of the octonion algebra. We then show how these representations by associative matrices allow a consistent octonionic gauge theory to be realized. We find that non-associativity implies the existence of new terms in the transformation laws of fields and the kinetic term of an octonionic Lagrangian. PACS numbers: 11.30.Ly, 12.10.Dm, 12.40.-y. Typeset Using REVTEX 1 L INTRODUCTION The aim of this work is to genuinely gauge the octonion algebra as opposed to relating properties of this algebra back to the well known theory of Lie Groups and fibre bundles. Typically most attempts to utilise the octonion symmetry in physics have revolved around considerations of the automorphism group G2 of the octonions and Jordan matrix representations of the octonions [1]. Our approach is more simple since we provide a spinorial approach to the octonion symmetry. Previous to this work there were already several indications that this should be possible. To begin with the statement of the gauge principle itself uno theory shall depend on the labelling of the internal symmetry space coordinates" seems to be independent of the exact nature of the gauge algebra and so should apply equally to non-associative algebras. The octonion algebra is an alternative algebra (the associator {x-1,y,i} = 0 always) X -1 so that the transformation law for a gauge field TM —• T^, = UY^U~ — ^(c^C/)(/ is well defined for octonionic transformations U. -
An Invitation to Local Fields
An Invitation to Local Fields Groups, Rings and Group Rings Ubatuba-S˜ao Paulo, 2008 Eduardo Tengan (ICMC-USP) “To get a book from these texts, only scissors and glue were needed.” J.-P. Serre, in response to receiving the 1995 Steele Prize for his book “Cours d’Arithm´etique” Copyright c 2008 E. Tengan Permission is granted to make and distribute verbatim copies of this document provided the copyright notice and this permission notice are preserved on all copies. The author was supported by FAPESP grant 06/59613-8. Preface 1 What is a Local Field? Historically, the first local field, the field of p-adic numbers Qp, was introduced in 1897 by Kurt Hensel, in an attempt to borrow ideas and techniques of power series in order to solve problems in Number Theory. Since its inception, local fields have attracted the attention of several mathematicians, and have found innumerable applications not only to Number Theory but also to Representation Theory, Division Algebras, Quadratic Forms and Algebraic Geometry. As a result, local fields are now consolidated as part of the standard repertoire of contemporary Mathematics. But what exactly is a local field? Local field is the name given to any finite field extension of either the field of p-adic numbers Qp or the field of Laurent power series Fp((t)). Local fields are complete topological fields, and as such are not too distant relatives of R and C. Unlike Q or Fp(t) (which are global fields), local fields admit a single valuation, hence the tag ‘local’. Local fields usually pop up as completions of a global field (with respect to one of the valuations of the latter). -
O/Zp OCTONION ALGEBRA and ITS MATRIX REPRESENTATIONS
Palestine Journal of Mathematics Vol. 6(1)(2017) , 307313 © Palestine Polytechnic University-PPU 2017 O Z OCTONION ALGEBRA AND ITS MATRIX = p REPRESENTATIONS Serpil HALICI and Adnan KARATA¸S Communicated by Ayman Badawi MSC 2010 Classications: Primary 17A20; Secondary 16G99. Keywords and phrases: Quaternion algebra, Octonion algebra, Cayley-Dickson process, Matrix representation. Abstract. Since O is a non-associative algebra over R, this real division algebra can not be algebraically isomorphic to any matrix algebras over the real number eld R. In this study using H with Cayley-Dickson process we obtain octonion algebra. Firstly, We investigate octonion algebra over Zp. Then, we use the left and right matrix representations of H to construct repre- sentation for octonion algebra. Furthermore, we get the matrix representations of O Z with the = p help of Cayley-Dickson process. 1 Introduction Let us summarize the notations needed to understand octonionic algebra. There are only four normed division algebras R, C, H and O [2], [6]. The octonion algebra is a non-commutative, non-associative but alternative algebra which discovered in 1843 by John T. Graves. Octonions have many applications in quantum logic, special relativity, supersymmetry, etc. Due to the non-associativity, representing octonions by matrices seems impossible. Nevertheless, one can overcome these problems by introducing left (or right) octonionic operators and xing the direc- tion of action. In this study we investigate matrix representations of octonion division algebra O Z . Now, lets = p start with the quaternion algebra H to construct the octonion algebra O. It is well known that any octonion a can be written by the Cayley-Dickson process as follows [7]. -
Local Fields
Part III | Local Fields Based on lectures by H. C. Johansson Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. The p-adic numbers Qp (where p is any prime) were invented by Hensel in the late 19th century, with a view to introduce function-theoretic methods into number theory. They are formed by completing Q with respect to the p-adic absolute value j − jp , defined −n n for non-zero x 2 Q by jxjp = p , where x = p a=b with a; b; n 2 Z and a and b are coprime to p. The p-adic absolute value allows one to study congruences modulo all powers of p simultaneously, using analytic methods. The concept of a local field is an abstraction of the field Qp, and the theory involves an interesting blend of algebra and analysis. Local fields provide a natural tool to attack many number-theoretic problems, and they are ubiquitous in modern algebraic number theory and arithmetic geometry. Topics likely to be covered include: The p-adic numbers. Local fields and their structure. Finite extensions, Galois theory and basic ramification theory. Polynomial equations; Hensel's Lemma, Newton polygons. Continuous functions on the p-adic integers, Mahler's Theorem. Local class field theory (time permitting). Pre-requisites Basic algebra, including Galois theory, and basic concepts from point set topology and metric spaces.