CLASS GROUPS AND CLASS NUMBERS

Transcription

1 CLASS GROUPS AND CLASS NUMBERS A THESIS SUBMITTED TO THE DEPARTMENT OF MATHEMATICS OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE WITH HONORS Carl W. Erickson May 2007

2 c Copyright by Carl W. Erickson 2007 All Rights Reserved ii

3 Acknowledgements I would like to thank my various advisors, principally Ben Brubaker, Allison Pacelli, Ken Ono, and Ravi Vakil. iii

4 Contents Acknowledgements iii 1 Introduction 1 2 Quadratic Forms Representations The Discriminant Finiteness of the Class Number The Class Group and Primes of the Form x 2 + ny Non-binary Quadratic Forms Algebraic Number Theory Introduction Central Concepts The Three Main Theorems Primes Class Field Theory Central Questions on Class Groups rank of Class Groups of Quadratic Fields Recent Discoveries Modular Forms Introduction Complex Multiplication iv

5 4.3 Borcherds Work p-adic Class Number Formulas Galois Representations 49 Bibliography 50 v

6 Chapter 1 Introduction My purpose in this thesis is to provide the reader with a taste of the principal theories connected with my number theoretic research. This has several benefits: first of all, it provides a single context for comprehending the theorems that I have proved in the past, as well as one proven here for the first time. I hope that this effort will be helpful to the reader. The project also gives me the chance and the challenge of properly comprehending the context around these theorems. I hope that the perspectives that I gain in piecing together this document will assist me as I continue to study number theory. In this brief introduction, I will give an overview of these theories, which at the same time serves as a summary of the organization of this thesis. A major theme in my work have been class groups and class numbers of number fields, as the title suggests, so I will begin with the earliest historical contact with these concepts. The study of class groups dates back to Gauss investigation of integral binary quadratic forms. These quadratic forms Q(x, y) = ax 2 + bxy + cy 2, a, b, c Z (1.1) and the theory describing them are connected to many of the principal notions in algebraic number theory. For instance, asking the question, If Q(x, y) is my favorite quadratic form and m is my favorite integer, do there exist integers x, y such that m = f(x, y)? connects to class field theory and the theory of complex multiplication. 1

7 CHAPTER 1. INTRODUCTION 2 Specifically, the orbits under a certain group action of a set of quadratic forms of a fixed discriminant are the classes of class groups, class numbers, and class fields. The basic theory of quadratic forms is thus an elemetary yet fascinating introduction to more advanced topics that figure into this thesis. The first chapter of this thesis will serve as a compact introduction to the theory of quadratic forms. It is my intention for this chapter to be accessible to middle level undergraduates. While it was the context of quadratic forms where class groups were first, implicitly studied, they were only understood for what they are in the context of algebraic number theory. The most basic kind of class group, called the ideal class group, is composed of the various structures of ideals within a number field. In the second chapter we develop the notion of a class group together with basic ideas of class field theory. I will discuss some major ideas and problems surrounding class groups. To conclude the chapter, I will prove a theorem constructing quadratic fields with special class group structure. The third chapter will discuss modular forms and their importance in number theory. After a general introduction we will naturally proceed to connections, mostly through the theory of complex multiplication, with previously introduced notions surrounding class groups. This chapter ends with a theorem that illuminates a small part of some important work of Borcherds with interesting consequences for class numbers of imaginary quadratic fields. We conclude with a theorem, appearing only in this thesis, that applies the most basic parts of the theory of Galois representations to ideas surrounding the theorem in the third chapter, resulting in a theorem that gives conditional lower bounds on class numbers of imaginary quadratic fields.

8 Chapter 2 Quadratic Forms 2.1 Representations Lagrange initiated the study of binary quadratic forms. A integral binary quadratic form is a polynomial Q(x, y) = ax 2 + bxy + cy 2 Z[x, y]. (2.1) The term integral refers to the fact that a, b, c Z and that we generally consider only x, y Z, and the term binary refers to the fact that there are exactly two variables, x and y. While the study of integral quadratic forms of more than two variables certainly plays an important role in number theory, which we will touch on in the final section of this chapter, we will simply say quadratic form to refer to integral binary quadratic forms, and we denote the set of quadratic forms Q. Mathematicians were first interested in studying quadratic forms because of their interest in what we now call the integers that a form represents. Definition We say that a quadratic form Q(x, y) represents an integer m if there exist integers x, y such that m = Q(x, y). When encountering a theory that studies a certain set of objects, it is common to ask why we care about the theory and its objects. The answer to this question in the case of quadratic forms is contained in the terminology itself: for example, one says 3

9 CHAPTER 2. QUADRATIC FORMS 4 that 13 is of the form x 2 + y 2. Thus we call Q(x, y) = x 2 + y 2 a quadratic form. This nuance to the terminology reminds us of exactly what we are studying - and it also indicates the paradigm shift (studying the integers to studying the forms themselves as objects) that allowed the advances that Legendre et. al. made. We are studying, in current terminology, the question of representing integers with various quadratic forms. For example, Fermat found that for any prime number p 1 (mod 4), there exist integers x, y such that x 2 + y 2 = p. David Cox s book Primes of the Form x 2 + ny 2 [3] uses the question implied in its title as an inroad to introducing his reader to many of the topics covered in this thesis, for example, class field theory and complex multiplication. I gratefully acknowledge the influence of his book on this thesis, and the organization for this section. We now formalize the main definitions surrounding our central question of representations of integers by quadratic forms. Definition Call a quadratic form Q(x, y) = ax 2 + bxy + cy 2 primitive if (a, b, c) = 1, i.e. if they share no common prime factor. Likewise, say that a quadratic form Q(x, y) properly represents an integer m if there exist integers x, y such that (x, y) = 1 and m = Q(x, y). It is natural to restrict our investigation of integer representations to primitive forms, because a nonprimitive form is simply an integer multiple of a primitive form. That is, if a quadratic form Q(x, y) is equal to np (x, y) where P (x, y) is a primitive quadratic form, then Q represents m Z if and only if n m and P represents n/m. Therefore questions of representing integers reduce to the primitive form case. The concept of proper representation likewise compactifies our investigation of representations. Now we understand our motivation for studying the set of quadratic forms, we may ask many questions that mathematicians ask about any set of objects. Among them are: 1. What are their relevant invariants? 2. Are there natural equivalence classes?

10 CHAPTER 2. QUADRATIC FORMS 5 3. Can we endow them with an algebraic structure? (group, module, etc.) There are pleasant answers to these questions, all closely related to the central question of representing integers with a quadratic form. Here we define the sense of equivalence we will use between quadratic forms. Definition A two quadratic forms Q(x, y), P (x, y) are properly equivalent if there is a matrix ( a c d b ) with integer entries and determinant 1 such that Q(x, y) = P (ax + by, cx + dy). An equivalence class under this equivalence relation is called a class. From the definition alone, it is not clear exactly why we should accept that this is a useful equivalence relation. The key motivation for this definition is that two quadratic forms properly represent the same integers if they are properly equivalent. The converse is not true, and leads us back into more history of the terms. Lagrange first defined a weaker sense of equivalence as follows. Definition A two quadratic forms Q(x, y), P (x, y) are equivalent if there is a matrix ( a c d b ) with integer entries and determinant 1 or -1 such that Q(x, y) = P (ax + by, cx + dy). An equivalence class under this equivalence relation is called a Lagrangian class. We will prove later, for many quadratic forms, the converse of the above statement. Precisely, we will prove that two positive definite quadratic forms properly represent the same integers if and only if they are equivalent. Though the weaker sense of equivalence is therefore useful for classifying representations of integers, we will see in the advanced theory that proper equivalence has greater importance. To begin investigating these equivalences, we will show that the classes are orbits of a group actions. We claim that the matrix group SL(2, Z) of integer matricies with determinant 1 acts on the set of quadratic forms as shown in the definition. It you are not familiar with SL(2, Z), verify for yourself that it is a group and compute, for example, the inverse of ( a c d b ) SL(2, Z). Likewise, one can verify that the group of matrices GL(2, Z) of matrices with integer entries and determinant 1 or 1 is a group.

11 CHAPTER 2. QUADRATIC FORMS 6 Proposition Given a matrix ( p q r s ) SL(2, Z), the mapping on Q defined by ( ) a b : Q(x, y) Q(ax + by, cx + dy) (2.2) c d defines a group action of SL(2, Z) or GL(2, Z) on Q. The proof that equation (2.2) defines a group action is rather straightforward once we develop a convenient way of representing quadratic forms as matrices and then making a way for SL(2, Z) to act on Q in terms of matrix multiplication. The representation contained in the easily verified equality ax 2 + bxy + cy 2 = [ x ] [ ] [ ] a b 2 x y, (2.3) b 2 y 2 where matrix multiplication is carried out on the right side of the equality. We will formalize this notion by writing M(Q(x, y)) for the symmetric matrix indicated in equation (2.3). The inverse function of M will be used on symmetric matrices arising from quadratic forms as well; for example, ([ ]) 1 2 M 1 = x 2 + 4xy + 9y 2. (2.4) 2 9 The proof of proposition relies on a matrix conjugation that gives the action on the matrices M(Q) we desire. Proof. Let Q(x, y) = ax 2 + bxy + cy 2. We will prove the proposition for the group GL(2, Z), and one can easily verify that the arguments function identically for SL(2, Z). We claim that the group action defined in equation (2.2) above satisfies ( ) p q Q(x, y) = r s [ x ] [ ] [ ] [ ] [ ] p r a b 2 p q x y, (2.5) q s r s y b 2 2

12 CHAPTER 2. QUADRATIC FORMS 7 or, equivalently, γ Q = [ x ] y γ T M(Q)γ [ ] x. (2.6) y for γ GL(2, Z) and Q Q. This claim follows simply from noting that [ x ] [ ] p r y = q s [ px + qy ] rx + sy, (2.7) and likewise [ p q r s ] [ x ] y = [ px + qy ] rx + sy. (2.8) Therefore we have that ( ) p q Q(x, y) = r s [ px + qy ] [ ] [ ] a b 2 px + qy rx + sy = Q(px+qy, rx+sy) (2.9) b 2 rx + sy 2 as desired. We leave it to the reader to verify that the group action axioms are satisfied, but this is almost immediate by using properties of matrix multiplication: one shows that γ 1 (γ 2 Q) = (γ 1 γ 2 ) Q for all γ 1, γ 2 SL(2, Z), and easily notes that ( ) Q = Q. With new information from the proof of proposition 2.1.1, we can prove the claim that motivated this definition of equivalence. Theorem If two quadratic forms are equivalent, then they properly represent the same integers. Proof. First we assume that Q(x, y) is equivalent to P (x, y). Thus there exists ( p q r s ) GL(2, Z) such that Q(x, y) = P (px+qy, rx+sy). It follows that P properly represents all integers properly represented by Q as long as (x, y) = 1 implies that (px +qy, rx + sy) = 1. This follows quickly on noting that [ p q r s ] 1 [ px + qy ] rx + sy = [ x ] y, (2.10)

13 CHAPTER 2. QUADRATIC FORMS 8 because a common factor of px + qy and rx + sy would divide the components of the vector γ [ px+qy rx+sy ] for γ GL(2, Z). Repeating the same argument with the role of P and Q exchanged completes the proof. The converse of this theorem is true (check this *), but for simplicity we will prove it later, and only in a certain case. The finer distinction of proper equivalence becomes evident in this proposition. Proposition A quadratic form Q(x, y) properly represents an integer m if and only if Q(x, y) is properly equivalent to a quadratic form mx 2 + bxy + cy 2 for some b, c Z. Proof. If Q(x, y) = mx 2 + bxy + cy 2, then Q(1, 0) = m is a proper representation of m by Q. For the converse, we assume that Q(x, y) properly represents m, say Q(p, q) = m. Then choose integers r, s as suggested immediately above. Note that ( p q r s ) SL(2, Z). Then note that the form P (x, y) := ( p q r s ) Q(x, y) satisfies P (1, 0) = Q(p, q) = m, so P (x, y) has the form mx 2 + bxy + cy 2 for some b, c Z. 2.2 The Discriminant With the sense of proper equivalence and its connection with representations in mind, we define the principal invariant of quadratic forms, the discriminant. Definition The discriminant of a quadratic form Q(x, y) = ax 2 + bxy + cy 2 is d = b 2 4ac. Note that the determinant of the matrix representation M(Q) of a quadratic form Q(x, y) = ax 2 + bxy + cy 2 is a b 2 b c 2 = ac b2 4, (2.11)

14 CHAPTER 2. QUADRATIC FORMS 9 a constant multiple of the discriminant of Q. Recalling that the action of γ SL(2, Z) on Q may be represented as γ Q = γ T M(Q)γ, we note that the determinant of γ T M(Q)γ is the same as that of M(Q). Thus we have proven this fact: Proposition The discriminant is invariant under the action of SL(2, Z) or GL(2, Z). Write Q d for the set of quadratic forms with discriminant d. Since the discriminant does not vary under these actions, we find this result. Corollary The group actions described in proposition restricts to a group action on Q d for each discriminant d. The discriminant plays a critical role as an invariant with respect to questions of representation. Most importantly, the sign of the discriminant strongly affects how we approach representations. For if Q(x, y) = ax 2 + bxy + cy 2 with discriminant d, we may write 4aQ(x, y) = (2ax + by) 2 dy 2. (2.12) It is immediate to see that the sign of the discriminant strongly influences the behavior of the quadratic form as x and y vary. But before discussing its sign, we investigate another ramifications for representations of integers suggested by equation (2.12). We consider modular arithmetic of representations with respect to d. We can begin by noting several properties of the discriminant with regards to modular arithmetic. First note that d 0 or 1 (mod 4) because d = b 2 4ac b 2 (mod 4). Thus the parity of b and d are always the same. When d = 0, then Q is the square of a linear form, so we will not consider 0 a discriminant. However, the rest of these possible values for the discriminant are taken on by some quadratic form. We will prove something stronger. Proposition (Cox Lemma 2.5) Let d 0, 1 (mod 4) be a nonzero integer and m be an odd integer relatively prime to d. Then m is properly represented by a primitive form of discriminant d if and only if d is a quadratic residue module m.

15 CHAPTER 2. QUADRATIC FORMS 10 Proof. If Q(x, y) properly represents m, then by proposition 2.1.1, we may assume that Q(x, y) = mx 2 + bxy + cy 2. Thus d = b 2 4mc, so d b 2 (mod m) is a square modulo m. Conversely, suppose that d b 2 (mod m). We may choose b with parity the same as d, since ***** By quadratic reciprocity, we can translate proposition into a statement about m modulo d. Therefore, given equivalence classes modulo d are represented by a quadratic form of discriminant d. Yet we cannot yet name a certain form with this property. Furthermore, we do not know which class(es) of quadratic forms with discrminant d represents m, nor even if there are finitely many proper equivalence classes. We will address these questions after restricting our investigations to forms with negative discriminant. When the discriminant is positive, it is not trival to show that there cannot be an infinite number of pairs x, y such that Q(x, y) = m for any integer m, positive or negative. This is the case because Q(x, y) according to equation (2.12) is the difference of two squares.*** On the other hand, when the discriminant d is negative, there are only finitely many x, y such that Q(x, y) is bounded.***(not really on the other hand ) Furthermore, all represented integers have the same sign when d is negative. The terminology is in this Definition A quadratic form Q with discriminant d is called positive definite if d < 0 and Q represents no negative integers; or negative definite if d < 0 and Q represents no positive integers. Otherwise, the form is called indefinite. Example we consider the form Q(x, y) = x 2 y 2 with positive discriminant equal to 4. The form Q clearly represents every odd integer since (2n) 2 (2n 1) 2 = 4n 1, and (2n + 1) 2 (2n) 2 = 4n + 1. And it represents every integer m divisible by 2 an even number of times, i.e. m = 2 2i m where m is odd and i is a positive integer, because if x 2 y 2 = m, then (2x) 2 (2y) 2 = 4m. However, it does not represent any integers divisible by 2 an odd number of times, since all integers squares are 0 or 1 modulo 4, and therefore x 2 y 2 is either 1, 0, or -1 modulo 4. This completes our discription of the values taken on by Q(x, y) = x 2 y 2.

16 CHAPTER 2. QUADRATIC FORMS 11 In short order one finds that the structure of representations of Q(x, y) = x 2 + y 2 is not as simple as those of x 2 y 2. Presently, we will focus on these positive definite forms - note that this simply demands that a, c > 0, so that studying negative definite forms is equivalent to studying positive definite forms. Though indefinite forms certain are important as well, but we will develop the context to understand them later (***make sure you do that) Our next object is to prove the finiteness of the number of classes of positive definite quadratic forms of a given discriminant. That is, 0 > d 0, 1 (mod 4), we want to show that there are finitely many orbits in Q d under the action of SL(2, Z). Usually this is done by introducing the concept of reduced forms and following the logic detailed here. A critical and useful notion that we have available with positive definite forms is Lagrange s reduced form. Definition A primitive positive definite form Q(x, y) = ax 2 + bxy + cy 2 is called reduced if b a c, and b 0 if b = a or a = c. (2.13) The usefulness of this notion is contained largely in the following Proposition Every primitive positive definite form is properly equivalent to a unique reduced form. We then note that d = 4ac b 2 4a 2 a 2 = 3a 2, so that there are finitely many choices for a. There are then finitely many choices for b since b a, and finally, no more than one choice of c for each choice of a and b. Therefore there are finitely many reduced forms of a given negative discriminant, so we have proved the following theorem. Theorem There are finitely many orbits of SL(2, Z) in Q d. This finite number of orbits is the first example of our central object of study in this thesis: Definition The class number h(d) of a negative discriminant d is the number of classes of properly equivalent forms in Q d, i.e. h(d) = SL(2, Z)\Q d.

17 CHAPTER 2. QUADRATIC FORMS 12 Theorem 2.2 is the first example of what is, in its generalized form, a critical theorem of algebraic number theory. The outline we have presented above is not intended to shed light on class numbers, but merely to indicate the historical context of these theories. *here or somehwere put in actual examples of equivalence with reduced forms, and the representations What we have neglected to show above is the proof of Proposition Usually this is accomplished by directly studying the action of SL(2, Z) on the coffients of the form above. See for example [3], Theorem 2.8. We will prove the finiteness of the class number in a completely different manner, which will introduce some new theory useful for chapter three while still remaining relatively elementary. 2.3 Finiteness of the Class Number Let H represent the upper half plane of the complex plane C, i.e. H = {z C : Im(z) > 0}. (2.14) The reader familiar with complex analysis is aware of the importance of fractional linear transformations. Definition A fractional linear transformation is a function f(z) = az + b cz + d (2.15) where ( a c d b ) is an invertible complex valued matrix. These matrices form a group called GL(2, C). A central fact about these functions is that any analytic automorphisms of the Riemann sphere C { } is a fractional lineat transformation. When a fractional linear transformation fixes the real line, i.e. f(z) R for z R, then either H is fixes or H is mapped onto H. Reasoning through these conditions we see that the matrices associated to fractional linear transformations

18 CHAPTER 2. QUADRATIC FORMS 13 that are analytic automorphisms of H are scalar multiples of real matricies. Since scalar multiplication does not affect the action of GL(2, C) on C, we may simply take the automorphisms of H to be GL(2, R), the group of invertible, real-valued matrices. The theory of these linear transformations is rich, but we turn to the simplest discrete (under the standard topology) subgroup of GL(2, R), namely SL(2, Z), to connect the action of SL(2, Z) on H with its action on quadratic forms. Usually we will speak simply of the action of matrix groups on H, with the fact that the action is by fractional linear transforms as in equation (2.15) left implicit. The importance of choosing a discrete subgroup Γ of GL(2, R) to act on is that the quotient Γ\H will then be Hausdorff, and can be compactified to make a Riemann surface. Important intuition to understand this quotient comes from the fundamental domain for the action of SL(2, Z) on H. Definition A fundamental domain for the action of G on A C is a simply connected subset of C containing exactly one element from each orbit of the action of G on A. The following defines the standard fundamental domain. Proposition The set F H defined by F = { 1 2 Re(z) < 1 2 and z > 1} { z = 1 and 1 2 Re(z) 0}. (2.16) is a fundamental domain for the action of SL(2, Z) on H. Proof. Get this from Serre [15] or from last summer s notes. Choose some quadratic form Q = ax 2 + bxy + cy 2 Q d of negative discriminant d. Associate to Q the quadratic irrational number α Q which is the solution in H to Q(x, 1) = ax 2 + bx + c = 0. We are guaranteed that there are two solutions in C to Q(x, 1) = 0, and recalling the quadratic formula, we note that one of these roots is in the upper half plane, namely α Q = b + d. (2.17) 2a

19 CHAPTER 2. QUADRATIC FORMS 14 Proposition The actions of SL(2, Z) on Q + and H commute in the sense that for all γ SL(2, Z) and Q Q +. α (γ Q) = γ 1 α Q (2.18) Proof. Fix some positive definite quadratic form Q Q + and let α Q be the solution in H to Q(x, 1) = 0. Choose γ = ( p q r s ) SL(2, Z). We want to show that γ 1 α Q is a root of (γ Q)(x, 1) = 0. In light of equation (2.5), we may write (γ Q)(x, 1) = 0 equivalently as or [ ] [ p r x 1 q s [ px + q ] [ ] [ ] [ ] a b 2 p q x = 0, (2.19) r s 1 b 2 2 ] [ ] [ ] a b 2 px + q rx + s = 0. (2.20) b 2 rx + s 2 Since rx+s 0 for x H and Q(x, y) is homogenous of degree 2 (that is, Q(λx, λy) = λ 2 Q(x, y)), we may let λ = (rx + s) 1 and write equivalently that [ px+q rx+s 1 ] [ a b 2 b 2 2 ] [ px+q rx+s 1 One readily sees that this equation is equivalent to ] = 0. (2.21) Q(γ x, 1) = 0, (2.22) so the unique solution in H to this equation is γ x = α Q. Thus since γ is an automorphism of H, the unique solution in H to (γ Q)(x, 1) = 0 is x = γ 1 α Q as desired. Proposition A quadratic form Q Q + is reduced if and only if α Q F. Proof. Assume that Q(x, y) = ax 2 +bxy +cy 2 is reduced. Recall that this means that b a c, and b 0 if b = a or a = c.

20 CHAPTER 2. QUADRATIC FORMS 15 If the real part of α Q is b/2a, so clearly 1 Re(α 2 Q) < 1 if b < a and Re(α 2 Q) = 1 2 if b = a. We also have that d = b 2 4ac b 2 4a 2, so α Q = b2 + d 4a 2 b2 + b 2 4a 2 4a 2 = 1. There is equality only when a = c, and in this case the definition of reduced implies that b 0. Thus we have that α Q = Re(α Q ) 0 as desired. This completes the proof that α Q F. The converse follows rather easily by following the inequalities above with the converse implications. Now we can deduce Proposition as a corollary to these facts. We restate it here as a corollary for convenience. Corollary Every positive definite quadratic form is properly equivalent to a unique reduced form. Proof. Choose any Q Q d and associate to Q the quadratic irrational α Q. By Proposition 2.3.1, there is a unique α F such that γ α Q = α for some γ SL(2, Z). Choose such a γ, so that by Proposition we know that γ 1 Q is reduced. Therefore every positive definite quadratic form is properly equivalent to a reduced form. Uniqueness follows from the fact that **check on Serre fundamental domain proof to cover nontrivial isotropy case 2.4 The Class Group and Primes of the Form x 2 + ny 2 On the most basic level, given two binary quadratic forms, it would be helpful to have a method for generating a third. Gauss and Dirichlet both developed a way to compose two binary forms to produce a new one. We will give Dirichlet s composition law applied to the case of positive definite binary quadratic forms, but we should note that there are composition laws for indefinite forms.

21 CHAPTER 2. QUADRATIC FORMS 16 Dirichlet composition is first suggested by the following fact. Proposition (x 2 + ny 2 )(z 2 + nw 2 ) = (xz ± nyw) 2 + n(xw yz) 2 This is easy to very by simply expanding terms. This property of the form x 2 +ny 2 is what motivate Cox to focus his attention on the representation of prime numbers of the form x 2 + ny 2. But something similar can be done for other pairs of quadratic forms of the same discriminant. Dirichlet composition works as follows. Let f(x, y) = ax 2 + bxy + cy 2 and g(x, y) = a x 2 + 2b xy + c y 2 be primitive postive definite forms of discriminant d. We also require that gcd(a, a, (b + b ) ) = 1. 2 We can prove that there exists a unique integer B (mod 2aa ) satisfying, B b (mod 2a) B b (mod 2a ) B 2 d (mod 4aa ) Now we can define the Dirichlet Composition of f(x, y) and g(x, y) to be F (x, y) = aa x 2 + 2Bxy + (B2 d 2 ) 4aa y 2 Proposition Given f(x, y) and g(x, y) as defined above, the Dirichlet composition F (x, y) is a primitive positive definite form of discriminant d. *Conclude binary quadratic forms section by discussing Cox s book and the class number problem, briefly. 2.5 Non-binary Quadratic Forms The theory of quadratic forms in more than two variables has also been an interesting area of research. In fact, it was not neccesary to use the matrix representation of

22 CHAPTER 2. QUADRATIC FORMS 17 quadratic forms as we have done in this chapter, but this kind of representation is very useful when dealine with more than two variables. A general quadratic form in n variables can be written as rowvectortimesmatrixtimesvector (2.23) One of the most interesting results regarding general integral quadratic forms is the question of representations. The general question is, is there a simple sufficient condition for a quadratic form to represent all integers? Conway and Schneeberger found that the answer is yes, but their result was not published until Bhargava [1] found a simpler proof. Theorem If an integral quadratic form with integral matrix represents all positive integers up to 15, then it represents all positive integers. *Complete comments *Ramanujan s ternary quadratic form

23 Chapter 3 Algebraic Number Theory 3.1 Introduction Algebraic number theory is most basically the study of algebraic numbers. However, it also includes questions that fit better into the frame of the algebraic study of numbers, for example, finding rational solutions to polynomial equations. The name algebraic number theory is thus doubly appropriate. For this chapter, we assume a familiarity with algebraic number theory because of the sheer amount of exposition that would be required to build up the concepts, but will still introduce the concepts to put them in their historical context. At the end of this chapter, we will prove a result on class groups of quadratic number fields The development of major concepts in algebraic number theory can be traced in parallel with attempts to prove Fermat s Last Theorem. Let us briefly summarize a mid-nineteeth century development that shows this correspondence. First note that it suffices to prove Fermat s Last Theorem for all odd prime exponents. Mathematicians of the mid-nineteenth century realized that the equation x p +y p = z p could be written as an equality of ideals in the ring Z[ζ p ] in the form (z) p = (x + y)(x + ζ p y)(x + ζ 2 py) (x + ζ p 1 p ) (3.1) where ζ p is a primitive pth root of unity. Lamé thought that he could use this program 18

24 CHAPTER 3. ALGEBRAIC NUMBER THEORY 19 to prove Fermat s Last Theorem, but Kummer realized that this technique would not work when the ring Z[ζ p ] did not have unique factorization, for example, with p = 23 being the smallest such prime. Kummer, however, developed a way to carry this program out for many prime exponenets. It is not hard to show that the ideals (x + ζpy) j and (x + ζp k y) are either relatively prime for all j k or have greatest common divisor (ζ p 1). In the first case, these coprime ideals product is a pth power, they must each be pth powers themselves. If the class number of Z[ζ] is not divisible by p, then a pth power of a non-principal ideal remains non-principal. Therefore, in this case, the ideal a such that a p = (x + ζpy) j must be principal. Arguing from here, and also covering the case that the factors (x + ζpy) j are not coprime (which is slightly harder), we can prove Fermat s Last Theorem for the exponent p as long as p does not divide the class number of Z[ζ p ]. This is what Kummer accomplished. Unfortunately, however, we do not know if primes p such that p does not divide the class number of Z[ζ p ], called regular, are infinite in number or not. The smallest odd prime that is irregular is 37, and we do know that there are infinitely many irregular primes. Therefore Kummer s strategy cannot yield a solution to Fermat s Last Theorem, though it does so in a conjecturally infinite number of cases. For more information, see [17]. If the reader is unfamiliar with algebraic number theory, then the above statements are not meant to be comprehensible. We have assumed that ideals factor uniquely in the ring Z[ζ p ] with justification, we and have not defined the class number of such a ring and shown how it relates to powers of ideals. Kummer certainly struggled to grasp these ideas much more than we will here, for he lacked our modern tools. We will show below that within rings of integers like Z[ζ p ], ideals factor uniquely into prime ideals, and we will define the class group to be (more or less) the quotient of all ideals by the principal ideals. This is why a pth power of a non-principal ideal would be non-principal if the class number was not divisible by p. Kummer had access to none of these ideas. He made his arguments with respect to elements of Z[ζ p ] instead of its ideals. This was very difficult for his purposes, because Z[ζ p ] does not have unique factorization into primes in general. We will see below that this is the case if and only if its class number is one (i.e. if and only if Z[ζ p ] is a principal

25 CHAPTER 3. ALGEBRAIC NUMBER THEORY 20 ideal domain, for then the quotient of all ideals by the principal ideals is trivial). Kummer formulated ideal numbers to get around the lack of unique factorizations. These numbers were not elements of Z[ζ p ], but allowed him to recover arithmetic structure lost because of the nonuniqueness of factorization in Z[ζ p ]. Later, Dedekind took Kummer s ideal numbers and made them the ideals of rings (which shows the origin of the name ideal - a name which is, on reflection, rather odd). In the rest of this chapter we will overview the central results and concepts of algebraic number theory and state some open questions before proving a construction for quadratic number fields with special class groups. 3.2 Central Concepts An algebraic number field, or simply number field, is a field K such that K Q and K is a finite dimension vector space over Q. We call this a finite extension of Q. Some authors may define a number field to be a subfield of C that is a finite extension of Q, but this additional structure is not necessary to speak of the number field. Moreover, to speak about properties of number fields, it is useful to fix an algebraic closure Q of Q in order. Such algebraic closures are not unique, though they are the same up to field isomorphism. One may show that algebraic number fields consist of algebraic numbers, numbers which are the roots of some polynomial with rational cofficients. Thus if we work in C, Q = {z C : f(z) = 0 for some f Q[x]}. (3.2) Clearly if α C is an algebraic number, then the set Q[α] of polynomials in α is a number field with basis {1, α,..., α n 1 } over Q for some n. Usually we write this field Q(α), rational functions in α, which amounts to the same thing. In fact, there is a unique polynomial f α (x) Q[x] that is monic and has minimal degree such that f α (α) = 0. This polynomial is called the minimal polynomial of α. For example, f i (x) = x and f 1+ 5 = x 2 + x 1. One can show that if g(x) Q[x] satisfies 2 g(α) = 0, then f(x) g(x). With these facts in mind, one can see that n, the size

26 CHAPTER 3. ALGEBRAIC NUMBER THEORY 21 of the basis of Q(α) over α, satisfies n = deg f α. We call this n the degree of the field extension Q(α)/Q and usuall write n = [Q(α) : Q]. In fact, any number field K can be written as K = Q(α) for some α Q. This statement is called the primitive element theorem. The facts above allow us to do Galois theory and study isomorphisms of fields. We may first suggest an isomorphism by noting that for any number field K we may write K = Q(α), from which it follows that K = Q[x]/(f α (x)). (3.3) But this is true for other roots β, γ,... of f α as well. Therefore, for example, the fields Q( 3 2) and Q(ρ 3 2) are isomorphic, where ρ is a primitive third root of unity. Taking another perspective on these isomorphisms, we may note that if n = [K : Q], then there are n embeddings of K into Q. Note that if we take the roots α 1,..., α n of a polynomial f(x) Q, then the field K = Q(α 1,..., α n ) is fixed by all possible embeddings into Q. In this case, we call the extension K/Q normal, and since all number field extensions satisfy another condition called separability, we can call the extension Galois. This term also applies to relative field extenions, i.e. K/L when the base field L is not the rational number field Q. All of the facts mentioned in these statements apply just as well to relative Galois extensions as to extensions of Q. Since f(x) will factor or split completely into linear factors in K[x] but not for any proper subfield of K, we also call K a splitting field over L. In the context of number fields, a field extension is Galois if and only if it is a splitting field. The set of embeddings of K into Q then form a group of automorphisms of K called its Galois group, denoted Gal(K/Q). Likewise, if K/L is Galois, then the set of embeddings of K into Q fixing L for the group Gal(K/L). Whether or not K/L is Galois, there are [K : L] such embeddigns. When a field extension K/L is not Galois, it is clear from what has been said above that K is contained in some Galois extension of L, for if K = L(α) we may simply adjoin the rest of the roots of f α L[x] to K. In fact, this field is the smallest field that is Galois over L and contains K, and therefore is called the Galois closure of K over L.

27 CHAPTER 3. ALGEBRAIC NUMBER THEORY 22 From now on we will take a less algebraically satsifying but simpler approach of considering all number fields as subsets of C. In other words, we have already fixed an algebraic closure Q of Q, as well as embeddings K Q C for each number field K. Then we consider Galois isomorphisms of number fields as restrictions of elements of the absolute Galois group Gal( Q/Q) on Q. The fundamental theorem of Galois theory states that the intermediate fields of the Galois extension K/L are in bijective correspondence with subgroups of the Galois group Gal(K/L). Moreover, this correspondence preserves the lattice structure of the group, because it is given by {Subgroups H of Gal(K/L)} Fixed field M of H {σ Gal(K/L) fixing M} {Intermediate fields of K/L} (3.4) Thus we can reduce some questions in field theory to group theory. The structures from Galois theory, namely the Galois group and Galois automorphisms of Galois extensions, are fundamental to answering many number theoretic questions. For example, it is helpful to know the Kronecker-Weber theorem: any finite abelian extension of Q, i.e. a Galois extension K/Q where Gal(K/Q) is finite abelian, is containted in a cyclotomic field Q(ζ l ) for some l Z + (see [9], p. 198). Also, we will see later that Galois theory in finite fields is very important to studying number fields. In fact, all that we have said above can be developed for abstract fields given some modest restrictions: if the field has characteristic 0 (i.e. it contains a field isomorphic to Q) or has characteristic p (i.e. it contains a field isomorphic to the finite field with p elements F p ) and every element is a pth power. Such fields are called perfect. ***Reference to further field theory In the example regarding Fermat s Last Theorem above, we saw how algebraic number theory can arise in attempting to solve a question about integers. Namely, looking for solutions to x p + y p = z p in Z[ζ p ] allows for insights not readily available in Z. However, we can also proceed to ask about number theoretic questions about integers in number fields, questions that are analogous to the questions in Z. Of course, these topics are connected - Fermat s Last Theorem connects to the question

28 CHAPTER 3. ALGEBRAIC NUMBER THEORY 23 of uniqueness of factorizations in Z[ζ p ]. Both are algebraic number theoretic. To begin doing number theory in number fields, we need a sense of integrality that classifies algebraic numbers. That is, we need to be able to complete the analogy Q is to Z as Q( 5) is to...? for any number field K in the place of Q( 5). We say that an element α Q is an integer provided that it is the root of a monic polynomial f(x) Z[x]. Though it is not trivial to prove, one may show that the set of algebraic integers forms a subring of Q. Then the intersection of a number field K with the set of algebraic integers forms a subring of is the ring of integers of K, denoted O K. However, it is not easy to see in general what a good Z basis or Z-generator for this ring is. Common examples are that for squarefree (positive or negative) integer d we have Z[ d] if d 2, 3 (mod 4) O Q( d) = [ ] Z if d 1 (mod 4), (3.5) 1+ d 2 and that the ring of integers of a cyclotomic field Q(ζ p ) is Z[ζ p ]. The ring of integers O K is integrally closed, that is, it contains any element in its field of fractions K which satisfies a monic polynomial in O K [x]. Equivallently, we may state that it the maximal order of K and is the integral closure of Z in K. This ring of integers is the object of number theoretic study in general number fields. Now we will exhibit the three most basic theorems of algebraic number theory, which classify these rings. Once it was well known that unique factorization into primes does not necessarily exist in rings of integers of number fields as it does in Z, other ways of classifying the arithmetic of the rings were sought. The results of this work created much more general important structures in mathematics, such as Dedekind s ideal theory. today s terms, the pleasant algebraic property of these rings is that they are Dedekind domains. If a ring R is a Dedekind domain, 1 this means that 1. R is Noetherian; 2. Every nonzero prime ideal of R is maximal; and 1 Here, rings are commutative with unity. In

29 CHAPTER 3. ALGEBRAIC NUMBER THEORY R is integrally closed. As an example, we can see that Z is a Dedekind domain because it is a principal ideal domain and therefore Noetherian; its nonzero prime ideals are (p), which is clearly maximal as well; and it is integrally closed because the only solutions in Q to monic polynomials in Z[x] are clearly integers. It is accurate to view Dedekind domains as almost principal ideal domans. For example, any ideal in a Dedekind domain is generated by some two elements (*citation). In Dedekind domains, and therefore in rings of integers, ideals factor uniquely into nonzero prime ideals even though elements may not. This is a major helpful property of Dedekind domains. Though we could state this theorem in the context of typical ideals - called integral ideals when needed - it is useful to develop the concept of fractional ideals. An additive subgroup A of a commutative ring R is a fractional ideal provided that it is of the form aa where a is an integral ideal of R and a is in the field of fractions of R. One readily sees that the fractional ideals of Z are of the form az where a Q. The advantage of studying fractional ideals is that they are a group, namely the group generated by the semigroup of integral ideals. Theorem Choose a number field K. Then given any nonzero fractional ideal a of K, there is a factorization a = p e 1 1 p e 2 2 p en n, where e i Z \ {0}, of a into unique prime ideals that is unique up to ordering. This is the first of the three principal theorems of algebraic number theory. It provides us useful arithemetic structure that correspons to our intuition about the rational integers. The next theorem involves our principal object of study, the class group of a number field, and has to do with the structure of ideals as well. Call a fractional ideal of K a principal fractional ideal if it may be written ao K where a K. If we write I K for the set of fractional ideals and P K for the set of principal fractional ideals, one readily sees that the quotient Cl K := I K /P K, the ideal class group of K classifies all of the structures of ideals in O K. This sense of structure

30 CHAPTER 3. ALGEBRAIC NUMBER THEORY 25 is, loosely, how an ideal looks as a lattice over Z, up to scalar multiple. Another fact is helpful to us: though all principal ideal domains are unique factorization domains, Dedekind domains with unique factorization are principal ideal domains. Therefore we can say that K s class group is trivial if and only if integers factor uniquely in K. There is an important exact sequence connected to ideal class group, which has analogies in other areas of mathematics. Let the function ι : K I K denote principal fractional ideal construction, i.e. ι(a) = (a) = ao K. Note that (a) = O K if and only if a O K. We then have an exact sequence 1 O K K I K Cl K 1. (3.6) This is an important general sequence, which also appears in topology in the study of mapping class groups. 3.3 The Three Main Theorems Many questions may be asked about class groups of number fields. We will summarize these in the next section, along with noting the close connection - indeed, isomorphism - between class groups of quadratic forms and class groups of quadratic number fields. However, these questions, such as do there exist arbitrarily large class groups?, make sense only after one has proven that the class number, i.e. the order of the class group, is finite. This is the second of the three main theorems of algebraic number theory. The final theorem tells us that the group of units of O K has finite rank and gives an expression for this rank. Note that this implies that the group of units can be quite complicated in the rings O K. Of course the unit group is abelian, but in the integers Z, the units are simply Z = {1, 1}. This means that we will have to drop our intuition about what a big integer looks like. For example, is a unit in the ring of integers Z[ 2] of Q( 2). In fact, many obstructions to conjectured number theoretic results come from the possible abudance of units in the rings O K. While these last two theorems are quite different, they share a similar technique, which, surprisingly perhaps, is geometric in nature. To state the final theorem, we

31 CHAPTER 3. ALGEBRAIC NUMBER THEORY 26 need a piece of terminology regarding the embeddings of the number field K (recall that we are now thinking of number fields as subfields of C of finite dimension over Q) into C. Let n = [K : Q] and recall that there are n embeddings of K into Q, and by extension, into C. Call an embedding σ real if σ(k) R, and number these embeddings σ 1,..., σ r. Otherwise, call σ complex, and note that its complex conjugate σ is a distinct embedding of K into C. Thus, number the complex embeddings σ r+1,..., σ r+s, σ r+1,..., σ r+s and note that n = r + 2s. Now we can state the final main theorems of algebraic number theory. Theorem Let K be a number field. Then its ideal class group Cl K is a finite abelian group. Theorem (cite Janusz) Let K be a number field with r real embeddings and s pairs of complex embeddings. The group of units in the subring O K of K is the direct product of the free abelian group of rank r + s 1 with a finite cyclic group. That is, O K = Z/mZ Zr+s 1. (3.7) for some integer m. We will give a proof sketch for these theorems, but first must recall some basic concepts in algebraic number theory: the norm and trace of elements or ideals in relative field extensions, and the discrimiant of the number field (really, of the ring of integers). Really these concepts are prior to the three main theorems, so in light of the level of presentation of this chapter, we will move on assuming these concepts for expediency. Consider the function v : K R r+2s defined by v(x) = (σ 1 (x),..., σ r (x), Re(σ r+1 (x)), Im(σ r+1 (x)),..., Re(σ r+s (x)), Im(σ r+s (x)). (3.8)

32 CHAPTER 3. ALGEBRAIC NUMBER THEORY 27 One may prove without much hassle that the image of O K or an ideal of O K under v is a full lattice in R r+2s = R n, i.e. a discrete, additive subgroup of dimension n. In fact, the volume of the image of an ideal a under v, i.e. the Lebesgue measure of R n /v(a), is 2 s N (a) K (3.9) where N ( ) is the absolute norm on K and K is the discriminant of K (Ṫhen we can try to find a way to apply this theorem of Minkowski (which is very geometrically intuitive) to get convenient bounds: Proposition (cite Janusz) Let L be a full lattice in the vector space V of dimension n over R and let X be a bounded, centrally symmetric, convex subset of V. If volx > 2 n vol(l) then X contains a nonzero point of L. If we look at an integral ideal a projected into this lattice, then this proposition allows us to conclude that there exists elements of small norm a. nonzero a a such that N K/Q (a) n! n n There exists a ( ) s 4 N (a) K. (3.10) π From this we may find the Minkowski bound on the norm of the smallest integral ideal in one of the ideal classes. Theorem For ideal class A in Cl K, there is an integral ideal a A such that N (a) n! n n ( ) s 4 K. (3.11) π Proof. Choose any ideal b, possibly fractional, in A. Then choose any b b such that b 1 = bb 1 is an integral ideal in A 1. Let M denote the right side of equation (3.11). We know by equation (3.10) that there is an element a 0 in b 1 such that NK/Q (a) N (b1 ) 1 M. Therefore the norm of the integral ideal ab 1 1 equal to the left side of the above

33 CHAPTER 3. ALGEBRAIC NUMBER THEORY 28 equation, and therefore is less than M. This ideal is in the same class as, so we are done. We can prove the unit theorem by apply the same kind of geometric lattice ideas and Minkowski s theorem to the image of a mapping into K that captures multiplicative instead of additive structure. Define the function l : K R r+s by l(a) = (ln σ 1 (a),..., ln σ r (a), 2 ln σ r+1 (a),..., 2 ln σ r+s (a) ). (3.12) First one notes that the unit group of O K will be mapped into the r+s 1 dimensional subspace of R r+s consisting of vectors (x 1,..., x r+s ) satisfying x i = 0. Arguing from here we get the unit theorem. The infinite order generators of the unit group of O K are called fundamental units. An important consequence of the unit theorem is Corollary Any number field K has infinitely many units unless K = Q or K is imaginary quadratic. This is the case because n = r + 2s but the rank of the unit group is r + s 1. An imaginary quadratic field has one pair of complex embeddings, and the rationals have one real embedding, but any other number field has at least two embeddings up to complex conjugation. Finding these fundamental units is not easy. For example, the norm of an integer a + b d Z[ d] = O Q( d where d > 0 is squarefree and not 1 modulo 4, is a 2 db 2, and this integer is a unit when the norm is ±1. The equation ±1 = a 2 db 2 (3.13) is called Pell s equation. It s solvability is not trivial, but it follows from the unit theorem. An example of a fundamental unit is Z[ 2]. We can see that our usual sense of size is thrown off when there are fundamental units when we observe that (1 + 2) 8 = is also a unit. In fact, the presence and intractability of fundamental units is a central obstruction to proving facts about class numbers.