7 Quadratic Forms in N Variables
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
Quadratic Forms and Automorphic Forms
Quadratic Forms and Automorphic Forms Jonathan Hanke May 16, 2011 2 Contents 1 Background on Quadratic Forms 11 1.1 Notation and Conventions . 11 1.2 Definitions of Quadratic Forms . 11 1.3 Equivalence of Quadratic Forms . 13 1.4 Direct Sums and Scaling . 13 1.5 The Geometry of Quadratic Spaces . 14 1.6 Quadratic Forms over Local Fields . 16 1.7 The Geometry of Quadratic Lattices – Dual Lattices . 18 1.8 Quadratic Forms over Local (p-adic) Rings of Integers . 19 1.9 Local-Global Results for Quadratic forms . 20 1.10 The Neighbor Method . 22 1.10.1 Constructing p-neighbors . 22 2 Theta functions 25 2.1 Definitions and convergence . 25 2.2 Symmetries of the theta function . 26 2.3 Modular Forms . 28 2.4 Asymptotic Statements about rQ(m) ...................... 31 2.5 The circle method and Siegel’s Formula . 32 2.6 Mass Formulas . 34 2.7 An Example: The sum of 4 squares . 35 2.7.1 Canonical measures for local densities . 36 2.7.2 Computing β1(m) ............................ 36 2.7.3 Understanding βp(m) by counting . 37 2.7.4 Computing βp(m) for all primes p ................... 38 2.7.5 Computing rQ(m) for certain m ..................... 39 3 Quaternions and Clifford Algebras 41 3.1 Definitions . 41 3.2 The Clifford Algebra . 45 3 4 CONTENTS 3.3 Connecting algebra and geometry in the orthogonal group . 47 3.4 The Spin Group . 49 3.5 Spinor Equivalence . 52 4 The Theta Lifting 55 4.1 Classical to Adelic modular forms for GL2 .................. -
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. -
QUADRATIC FORMS and DEFINITE MATRICES 1.1. Definition of A
QUADRATIC FORMS AND DEFINITE MATRICES 1. DEFINITION AND CLASSIFICATION OF QUADRATIC FORMS 1.1. Definition of a quadratic form. Let A denote an n x n symmetric matrix with real entries and let x denote an n x 1 column vector. Then Q = x’Ax is said to be a quadratic form. Note that a11 ··· a1n . x1 Q = x´Ax =(x1...xn) . xn an1 ··· ann P a1ixi . =(x1,x2, ··· ,xn) . P anixi 2 (1) = a11x1 + a12x1x2 + ... + a1nx1xn 2 + a21x2x1 + a22x2 + ... + a2nx2xn + ... + ... + ... 2 + an1xnx1 + an2xnx2 + ... + annxn = Pi ≤ j aij xi xj For example, consider the matrix 12 A = 21 and the vector x. Q is given by 0 12x1 Q = x Ax =[x1 x2] 21 x2 x1 =[x1 +2x2 2 x1 + x2 ] x2 2 2 = x1 +2x1 x2 +2x1 x2 + x2 2 2 = x1 +4x1 x2 + x2 1.2. Classification of the quadratic form Q = x0Ax: A quadratic form is said to be: a: negative definite: Q<0 when x =06 b: negative semidefinite: Q ≤ 0 for all x and Q =0for some x =06 c: positive definite: Q>0 when x =06 d: positive semidefinite: Q ≥ 0 for all x and Q = 0 for some x =06 e: indefinite: Q>0 for some x and Q<0 for some other x Date: September 14, 2004. 1 2 QUADRATIC FORMS AND DEFINITE MATRICES Consider as an example the 3x3 diagonal matrix D below and a general 3 element vector x. 100 D = 020 004 The general quadratic form is given by 100 x1 0 Q = x Ax =[x1 x2 x3] 020 x2 004 x3 x1 =[x 2 x 4 x ] x2 1 2 3 x3 2 2 2 = x1 +2x2 +4x3 Note that for any real vector x =06 , that Q will be positive, because the square of any number is positive, the coefficients of the squared terms are positive and the sum of positive numbers is always positive. -
Symmetric Matrices and Quadratic Forms Quadratic Form
Symmetric Matrices and Quadratic Forms Quadratic form • Suppose 풙 is a column vector in ℝ푛, and 퐴 is a symmetric 푛 × 푛 matrix. • The term 풙푇퐴풙 is called a quadratic form. • The result of the quadratic form is a scalar. (1 × 푛)(푛 × 푛)(푛 × 1) • The quadratic form is also called a quadratic function 푄 풙 = 풙푇퐴풙. • The quadratic function’s input is the vector 푥 and the output is a scalar. 2 Quadratic form • Suppose 푥 is a vector in ℝ3, the quadratic form is: 푎11 푎12 푎13 푥1 푇 • 푄 풙 = 풙 퐴풙 = 푥1 푥2 푥3 푎21 푎22 푎23 푥2 푎31 푎32 푎33 푥3 2 2 2 • 푄 풙 = 푎11푥1 + 푎22푥2 + 푎33푥3 + ⋯ 푎12 + 푎21 푥1푥2 + 푎13 + 푎31 푥1푥3 + 푎23 + 푎32 푥2푥3 • Since 퐴 is symmetric 푎푖푗 = 푎푗푖, so: 2 2 2 • 푄 풙 = 푎11푥1 + 푎22푥2 + 푎33푥3 + 2푎12푥1푥2 + 2푎13푥1푥3 + 2푎23푥2푥3 3 Quadratic form • Example: find the quadratic polynomial for the following symmetric matrices: 1 −1 0 1 0 퐴 = , 퐵 = −1 2 1 0 2 0 1 −1 푇 1 0 푥1 2 2 • 푄 풙 = 풙 퐴풙 = 푥1 푥2 = 푥1 + 2푥2 0 2 푥2 1 −1 0 푥1 푇 2 2 2 • 푄 풙 = 풙 퐵풙 = 푥1 푥2 푥3 −1 2 1 푥2 = 푥1 + 2푥2 − 푥3 − 0 1 −1 푥3 2푥1푥2 + 2푥2푥3 4 Motivation for quadratic forms • Example: Consider the function 2 2 푄 푥 = 8푥1 − 4푥1푥2 + 5푥2 Determine whether Q(0,0) is the global minimum. • Solution we can rewrite following equation as quadratic form 8 −2 푄 푥 = 푥푇퐴푥 푤ℎ푒푟푒 퐴 = −2 5 The matrix A is symmetric by construction. -
Quadratic Forms ¨
MATH 355 Supplemental Notes Quadratic Forms Quadratic Forms Each of 2 x2 ?2x 7, 3x18 x11, and 0 ´ ` ´ ⇡ is a polynomial in the single variable x. But polynomials can involve more than one variable. For instance, 1 ?5x8 x x4 x3x and 3x2 2x x 7x2 1 ´ 3 1 2 ` 1 2 1 ` 1 2 ` 2 are polynomials in the two variables x1, x2; products between powers of variables in terms are permissible, but all exponents in such powers must be nonnegative integers to fit the classification polynomial. The degrees of the terms of 1 ?5x8 x x4 x3x 1 ´ 3 1 2 ` 1 2 are 8, 5 and 4, respectively. When all terms in a polynomial are of the same degree k, we call that polynomial a k-form. Thus, 3x2 2x x 7x2 1 ` 1 2 ` 2 is a 2-form (also known as a quadratic form) in two variables, while the dot product of a constant vector a and a vector x Rn of unknowns P a1 x1 »a2fi »x2fi a x a x a x a x “ . “ 1 1 ` 2 2 `¨¨¨` n n ¨ — . ffi ¨ — . ffi — ffi — ffi —a ffi —x ffi — nffi — nffi – fl – fl is a 1-form, or linear form in the n variables found in x. The quadratic form in variables x1, x2 T 2 2 x1 ab2 x1 ax1 bx1x2 cx2 { Ax, x , ` ` “ «x2ff «b 2 c ff«x2ff “ x y { for ab2 x1 A { and x . “ «b 2 c ff “ «x2ff { Similarly, a quadratic form in variables x1, x2, x3 like 2x2 3x x x2 4x x 5x x can be written as Ax, x , 1 ´ 1 2 ´ 2 ` 1 3 ` 2 3 x y where 2 1.52 x ´ 1 A 1.5 12.5 and x x . -
Quadratic Form - Wikipedia, the Free Encyclopedia
Quadratic form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Quadratic_form Quadratic form From Wikipedia, the free encyclopedia In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, is a quadratic form in the variables x and y. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form). Contents 1 Introduction 2 History 3 Real quadratic forms 4 Definitions 4.1 Quadratic spaces 4.2 Further definitions 5 Equivalence of forms 6 Geometric meaning 7 Integral quadratic forms 7.1 Historical use 7.2 Universal quadratic forms 8 See also 9 Notes 10 References 11 External links Introduction Quadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form: where a,…,f are the coefficients.[1] Note that quadratic functions, such as ax2 + bx + c in the one variable case, are not quadratic forms, as they are typically not homogeneous (unless b and c are both 0). The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the 1 of 8 27/03/2013 12:41 Quadratic form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Quadratic_form coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. -
Ellipse Axes, Eccentricity, and Direction of Rotation
Math 558 Prof. J. Rauch Ellipse Axes. Aspect ratio, and, Direction of Rotation for Planar Centers This handout concerns 2×2 constant coefficient real homogeneous linear systems X0 = AX in the case that A has a pair of complex conjugate eigenvalues a ± ib, b 6= 0. The orbits are elliptical if a = 0 while in the general case, e−atX(t) is elliptical. The latter curves are the solutions of the equation tr A X0 = (A − aI)X; a = : 2 For either elliptical or spiral orbits we associate this modified equation that has elliptical orbits. The new coefficient matrix A − (tr A=2)I has trace equal to zero and positive determinant. These two conditions characterize the matrices with a pair of non zero complex conjugate eigenvalues on the imaginary axis. Those are the matrices for which the phase portrait is a center. We show how to compute the axes of the ellipse, the eccentricity of the ellipse, and the direction of rotation, clockwise or counterclockwise. x1. Direction of rotation. To determine the direction of rotation it suffices to find the direction of the rotation on the positive x1-axis. If the flow is upward (resp. downward) then the swirl is counterclockwise (resp. clockwise). For a matrix A that has no real eigenvalues, the direction of swirl is counterclockwise if and only if the second coordinate of 1 a A = 11 0 a21 is positive, if and only if a21 > 0. The swirl is counterclockwise if and only if a21 > 0. Freeing this computation from the choice (1; 0) introduces an interesting real quadratic form. -
8.2 Quadratic Forms Example 1 Consider the Function Q(X1,X2)=8X 1
8.2 Quadratic Forms Example 1 Consider the function q(x ; x ) = 8x2 4x x + 5x2 1 2 1 − 1 2 2 Determine whether q(0; 0) is the global mini- mum. Solution based on matrix technique Rewrite x1 2 2 q( ) = 8x1 4x1x2 + 5x2 " x2 # − x 8x 2x = 1 1 − 2 x 2x + 5x " 2 # " − 1 2 # Note that we split the contribution 4x x − 1 2 equally among the two components. More succinctly, we can write 8 2 q(~x) = ~x A~x; where A = − · 2 5 " − # 1 or q(~x) = ~xT A~x The matrix A is symmetric by construction. By the spectral theorem, there is an orthonormal eigenbasis ~v1; ~v2 for A. We find 1 2 1 1 ~v1 = ; ~v2 = p5 1 p5 2 " − # " # with associated eigenvalues λ1 = 9 and λ2 = 4. Let ~x = c1~v1 + c2~v2, we can express the value of the function as follows: q(~x) = ~x A~x = (c ~v + c ~v ) (c λ ~v + c λ ~v ) · 1 1 2 2 · 1 1 1 2 2 2 2 2 2 2 = λ1c1 + λ2c2 = 9c1 + 4c2 Therefore, q(~x) > 0 for all nonzero ~x. q(0; 0) = 0 is the global minimum of the function. Def 8.2.1 Quadratic forms n A function q(x1; x2; : : : ; xn) from R to R is called a quadratic form if it is a linear combina- tion of functions of the form xixj. A quadratic form can be written as q(~x) = ~x A~x = ~xT A~x · for a symmetric n n matrix A. -
Math 604 Introduction to Quadratic Forms Fall 2016
Yale University Department of Mathematics Math 604 Introduction to Quadratic Forms Fall 2016 Final Exam (due 5:30 pm December 21) 1. Quadratic forms over finite fields. Let q be an odd prime power. × ×2 (a) Show that Fq =Fq has order 2. (b) Show than any element of Fq is a sum of two squares. (c) Show that every binary quadratic form over Fq represents every nonzero element. × ×2 (d) Show that if q ≡ 1 (mod 4) then W (Fq) is isomorphic to the ring Z=2Z[Fq =Fq ] (e) Show that if q ≡ 3 (mod 4) then W (Fq) is isomorphic to the ring Z=4Z. (f) Show that the isomorphic type of GW (Fq) as a ring does not depend on q. 2. Characteristic 2, scary! Let F be a field of characteristic 2 and a; b 2 F . A quadratic form q : V ! F is nondegenerate if its associated bilinear form bq : V × V ! F defined by bq(v; w) = q(v + w) − q(v) − q(w) has a radical of dimension at most 1. Define the 2 2 2 quadratic form [a; b] on F by (x; y) 7! ax + xy + by . Let H be the hyperbolic form on F 2 defined by (x; y) 7! xy. (a) Prove that hai is nondegenerate for any a 2 F × but that ha; bi is always degenerate. (b) Prove that [a; b] is nondegenerate for any a; b 2 F . (c) Prove that any nondegenerate quadratic form of dimension 2 over F is isometric to a binary quadratic form [a; b] for some a; b 2 F .