Computing the Hilbert Class Field of Real Quadratic Fields
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The Class Number One Problem for Imaginary Quadratic Fields
MODULAR CURVES AND THE CLASS NUMBER ONE PROBLEM JEREMY BOOHER Gauss found 9 imaginary quadratic fields with class number one, and in the early 19th century conjectured he had found all of them. It turns out he was correct, but it took until the mid 20th century to prove this. Theorem 1. Let K be an imaginary quadratic field whose ring of integers has class number one. Then K is one of p p p p p p p p Q(i); Q( −2); Q( −3); Q( −7); Q( −11); Q( −19); Q( −43); Q( −67); Q( −163): There are several approaches. Heegner [9] gave a proof in 1952 using the theory of modular functions and complex multiplication. It was dismissed since there were gaps in Heegner's paper and the work of Weber [18] on which it was based. In 1967 Stark gave a correct proof [16], and then noticed that Heegner's proof was essentially correct and in fact equiv- alent to his own. Also in 1967, Baker gave a proof using lower bounds for linear forms in logarithms [1]. Later, Serre [14] gave a new approach based on modular curve, reducing the class number + one problem to finding special points on the modular curve Xns(n). For certain values of n, it is feasible to find all of these points. He remarks that when \N = 24 An elliptic curve is obtained. This is the level considered in effect by Heegner." Serre says nothing more, and later writers only repeat this comment. This essay will present Heegner's argument, as modernized in Cox [7], then explain Serre's strategy. -
Introduction to Class Field Theory and Primes of the Form X + Ny
Introduction to Class Field Theory and Primes of the Form x2 + ny2 Che Li October 3, 2018 Abstract This paper introduces the basic theorems of class field theory, based on an exposition of some fundamental ideas in algebraic number theory (prime decomposition of ideals, ramification theory, Hilbert class field, and generalized ideal class group), to answer the question of which primes can be expressed in the form x2 + ny2 for integers x and y, for a given n. Contents 1 Number Fields1 1.1 Prime Decomposition of Ideals..........................1 1.2 Basic Ramification Theory.............................3 2 Quadratic Fields6 3 Class Field Theory7 3.1 Hilbert Class Field.................................7 3.2 p = x2 + ny2 for infinitely n’s (1)........................8 3.3 Example: p = x2 + 5y2 .............................. 11 3.4 Orders in Imaginary Quadratic Fields...................... 13 3.5 Theorems of Class Field Theory.......................... 16 3.6 p = x2 + ny2 for infinitely many n’s (2)..................... 18 3.7 Example: p = x2 + 27y2 .............................. 20 1 Number Fields 1.1 Prime Decomposition of Ideals We will review some basic facts from algebraic number theory, including Dedekind Domain, unique factorization of ideals, and ramification theory. To begin, we define an algebraic number field (or, simply, a number field) to be a finite field extension K of Q. The set of algebraic integers in K form a ring OK , which we call the ring of integers, i.e., OK is the set of all α 2 K which are roots of a monic integer polynomial. In general, OK is not a UFD but a Dedekind domain. -
Units and Primes in Quadratic Fields
Units and Primes 1 / 20 Overview Evolution of Primality Norms, Units, and Primes Factorization as Products of Primes Units in a Quadratic Field 2 / 20 Rational Integer Primes Definition A rational integer m is prime if it is not 0 or ±1, and possesses no factors but ±1 and ±m. 3 / 20 Division Property of Rational Primes Theorem 1.3 Let p; a; b be rational integers. If p is prime and and p j ab, then p j a or p j b. 4 / 20 Gaussian Integer Primes Definition Let π; α; β be Gaussian integers. We say that prime if it is not 0, not a unit, and if in every factorization π = αβ, one of α or β is a unit. Note A Gaussian integer is a unit if there exists some Gaussian integer η such that η = 1. 5 / 20 Division Property of Gaussian Integer Primes Theorem 1.7 Let π; α; β be Gaussian integers. If π is prime and π j αβ, then π j α or π j β. 6 / 20 Algebraic Integers Definition An algebraic number is an algebraic integer if its minimal polynomial over Q has only rational integers as coefficients. Question How does the notion of primality extend to the algebraic integers? 7 / 20 Algebraic Integer Primes Let A denote the ring of all algebraic integers, let K = Q(θ) be an algebraic extension, and let R = A \ K. Given α; β 2 R, write α j β when there exists some γ 2 R with αγ = β. Definition Say that 2 R is a unit in K when there exists some η 2 R with η = 1. -
A Concrete Example of Prime Behavior in Quadratic Fields
A CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS CASEY BRUCK 1. Abstract The goal of this paper is to provide a concise way for undergraduate math- ematics students to learn about how prime numbers behave in quadratic fields. This paper will provide students with some basic number theory background required to understand the material being presented. We start with the topic of quadratic fields, number fields of degree two. This section includes some basic properties of these fields and definitions which we will be using later on in the paper. The next section introduces the reader to prime numbers and how they are different from what is taught in earlier math courses, specifically the difference between an irreducible number and a prime number. We then move onto the majority of the discussion on prime numbers in quadratic fields and how they behave, specifically when a prime will ramify, split, or be inert. The final section of this paper will detail an explicit example of a quadratic field and what happens to prime numbers p within it. The specific field we choose is Q( −5) and we will be looking at what forms primes will have to be of for each of the three possible outcomes within the field. 2. Quadratic Fields One of the most important concepts of algebraic number theory comes from the factorization of primes in number fields. We want to construct Date: March 17, 2017. 1 2 CASEY BRUCK a way to observe the behavior of elements in a field extension, and while number fields in general may be a very complicated subject beyond the scope of this paper, we can fully analyze quadratic number fields. -
Computational Techniques in Quadratic Fields
Computational Techniques in Quadratic Fields by Michael John Jacobson, Jr. A thesis presented to the University of Manitoba in partial fulfilment of the requirements for the degree of Master of Science in Computer Science Winnipeg, Manitoba, Canada, 1995 c Michael John Jacobson, Jr. 1995 ii I hereby declare that I am the sole author of this thesis. I authorize the University of Manitoba to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Manitoba to reproduce this thesis by photocopy- ing or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. iii The University of Manitoba requires the signatures of all persons using or photocopy- ing this thesis. Please sign below, and give address and date. iv Abstract Since Kummer's work on Fermat's Last Theorem, algebraic number theory has been a subject of interest for many mathematicians. In particular, a great amount of effort has been expended on the simplest algebraic extensions of the rationals, quadratic fields. These are intimately linked to binary quadratic forms and have proven to be a good test- ing ground for algebraic number theorists because, although computing with ideals and field elements is relatively easy, there are still many unsolved and difficult problems re- maining. For example, it is not known whether there exist infinitely many real quadratic fields with class number one, and the best unconditional algorithm known for computing the class number has complexity O D1=2+ : In fact, the apparent difficulty of com- puting class numbers has given rise to cryptographic algorithms based on arithmetic in quadratic fields. -
Open Problems: CM-Types Acting on Class Groups
Open problems: CM-types acting on class groups Gabor Wiese∗ October 29, 2003 These are (modified) notes and slides of a talk given in the Leiden number theory seminar, whose general topic are Shimura varieties. The aim of the talk is to formulate a purely number theoretic question (of Brauer-Siegel type), whose (partial) answer might lead to progress on the so-called André-Oort conjecture on special subvarieties of Shimura varieties (thus the link to the seminar). The question ought to be considered as a special case of problem 14 in [EMO], which was proposed by Bas Edixhoven and which we recall now. Let g be a positive integer and let Ag be the moduli space of principally polarized abelian varieties of dimension g. Do there exist C; δ > 0 such that for every CM-point x of Ag (i.e. the corresponding abelian variety has CM) one has G :x C discr(R ) δ; j Q j ≥ j x j where Rx is the centre of the endomorphism ring of the abelian variety correspond- ing to x??? What we will call the Siegel question in the sequel, is obtained by restricting to simple abelian varieties. In the talk I will prove everything that seems to be known on the question up to this moment, namely the case of dimension 2. This was worked out by Bas Edixhoven in [E]. The arguments used are quite “ad hoc” and I will try to point out, where obstacles to gener- alisations seem to lie. The boxes correspond more or less to the slides I used. -
Background on Local Fields and Kummer Theory
MATH 776 BACKGROUND ON LOCAL FIELDS AND KUMMER THEORY ANDREW SNOWDEN Our goal at the moment is to prove the Kronecker{Weber theorem. Before getting to this, we review some of the basic theory of local fields and Kummer theory, both of which will be used constantly throughout this course. 1. Structure of local fields Let K=Qp be a finite extension. We denote the ring of integers by OK . It is a DVR. There is a unique maximal ideal m, which is principal; any generator is called a uniformizer. We often write π for a uniformizer. The quotient OK =m is a finite field, called the residue field; it is often denoted k and its cardinality is often denoted q. Fix a uniformizer π. Every non-zero element x of K can be written uniquely in the form n uπ where u is a unit of OK and n 2 Z; we call n the valuation of x, and often denote it S −n v(x). We thus have K = n≥0 π OK . This shows that K is a direct union of the fractional −n ideals π OK , each of which is a free OK -module of rank one. The additive group OK is d isomorphic to Zp, where d = [K : Qp]. n × ∼ × The decomposition x = uπ shows that K = Z × U, where U = OK is the unit group. This decomposition is non-canonical, as it depends on the choice of π. The exact sequence 0 ! U ! K× !v Z ! 0 is canonical. Choosing a uniformizer is equivalent to choosing a splitting of this exact sequence. -
Class Field Theory & Complex Multiplication
Class Field Theory & Complex Multiplication S´eminairede Math´ematiquesSup´erieures,CRM, Montr´eal June 23-July 4, 2014 Eknath Ghate 1 Introduction An elliptic curve has complex multiplication (or CM for short) if it has endo- morphisms other than the obvious ones given by multiplication by integers. The main purpose of these notes is to show that the j-invariant of an elliptic curve with CM along with its torsion points can be used to explicitly generate the maximal abelian extension of an imaginary quadratic field. This result is analogous to the Kronecker-Weber theorem which states that the maximal abelian extension of Q is generated by the values of the exponential function e2πix at the torsion points Q=Z of the group C=Z. The CM theory of elliptic curves is due to many authors, including Kro- necker, Weber, Hasse, Deuring, Shimura. Our exposition is based on Chap- ters 4 and 5 of Shimura [1], and Chapter 2 of Silverman [3]. For standard facts about elliptic curves we sometimes refer the reader to Silverman [2]. 2 What is complex multiplication? Let E and E0 be elliptic curves defined over an algebraically closed field k. A homomorphism λ : E ! E0 is a rational map that is also a group homomorphism. An isogeny λ : E ! E0 is a homomorphism with finite kernel. Denote the ring of all endomorphisms of E by End(E), and set EndQ(E) = End(E) ⊗ Q. If E is an elliptic curve defined over C, then E is isomorphic to C=L for a lattice L ⊂ C. -
20 the Kronecker-Weber Theorem
18.785 Number theory I Fall 2016 Lecture #20 11/17/2016 20 The Kronecker-Weber theorem In the previous lecture we established a relationship between finite groups of Dirichlet characters and subfields of cyclotomic fields. Specifically, we showed that there is a one-to- one-correspondence between finite groups H of primitive Dirichlet characters of conductor dividing m and subfields K of Q(ζm)=Q under which H can be viewed as the character group of the finite abelian group Gal(K=Q) and the Dedekind zeta function of K factors as Y ζK (x) = L(s; χ): χ2H Now suppose we are given an arbitrary finite abelian extension K=Q. Does the character group of Gal(K=Q) correspond to a group of Dirichlet characters, and can we then factor the Dedekind zeta function ζK (S) as a product of Dirichlet L-functions? The answer is yes! This is a consequence of the Kronecker-Weber theorem, which states that every finite abelian extension of Q lies in a cyclotomic field. This theorem was first stated in 1853 by Kronecker [2] and provided a partial proof for extensions of odd degree. Weber [6] published a proof 1886 that was believed to address the remaining cases; in fact Weber's proof contains some gaps (as noted in [4]), but in any case an alternative proof was given a few years later by Hilbert [1]. The proof we present here is adapted from [5, Ch. 14] 20.1 Local and global Kronecker-Weber theorems We now state the (global) Kronecker-Weber theorem. -
Algebraic Number Theory II
Algebraic number theory II Uwe Jannsen Contents 1 Infinite Galois theory2 2 Projective and inductive limits9 3 Cohomology of groups and pro-finite groups 15 4 Basics about modules and homological Algebra 21 5 Applications to group cohomology 31 6 Hilbert 90 and Kummer theory 41 7 Properties of group cohomology 48 8 Tate cohomology for finite groups 53 9 Cohomology of cyclic groups 56 10 The cup product 63 11 The corestriction 70 12 Local class field theory I 75 13 Three Theorems of Tate 80 14 Abstract class field theory 83 15 Local class field theory II 91 16 Local class field theory III 94 17 Global class field theory I 97 0 18 Global class field theory II 101 19 Global class field theory III 107 20 Global class field theory IV 112 1 Infinite Galois theory An algebraic field extension L/K is called Galois, if it is normal and separable. For this, L/K does not need to have finite degree. For example, for a finite field Fp with p elements (p a prime number), the algebraic closure Fp is Galois over Fp, and has infinite degree. We define in this general situation Definition 1.1 Let L/K be a Galois extension. Then the Galois group of L over K is defined as Gal(L/K) := AutK (L) = {σ : L → L | σ field automorphisms, σ(x) = x for all x ∈ K}. But the main theorem of Galois theory (correspondence between all subgroups of Gal(L/K) and all intermediate fields of L/K) only holds for finite extensions! To obtain the correct answer, one needs a topology on Gal(L/K): Definition 1.2 Let L/K be a Galois extension. -
Nonvanishing of Hecke L-Functions and Bloch-Kato P
NONVANISHING OF HECKE L{FUNCTIONS AND BLOCH-KATO p-SELMER GROUPS ARIANNA IANNUZZI, BYOUNG DU KIM, RIAD MASRI, ALEXANDER MATHERS, MARIA ROSS, AND WEI-LUN TSAI Abstract. The canonical Hecke characters in the sense of Rohrlich form a set of algebraic Hecke characters with important arithmetic properties. In this paper, we prove that for an asymptotic density of 100% of imaginary quadratic fields K within certain general families, the number of canonical Hecke characters of K whose L{function has a nonvanishing central value is jdisc(K)jδ for some absolute constant δ > 0. We then prove an analogous density result for the number of canonical Hecke characters of K whose associated Bloch-Kato p-Selmer group is finite. Among other things, our proofs rely on recent work of Ellenberg, Pierce, and Wood on bounds for torsion in class groups, and a careful study of the main conjecture of Iwasawa theory for imaginary quadratic fields. 1. Introduction and statement of results p Let K = Q( −D) be an imaginary quadratic field of discriminant −D with D > 3 and D ≡ 3 mod 4. Let OK be the ring of integers and letp "(n) = (−D=n) = (n=D) be the Kronecker symbol. × We view " as a quadratic character of (OK = −DOK ) via the isomorphism p ∼ Z=DZ = OK = −DOK : p A canonical Hecke character of K is a Hecke character k of conductor −DOK satisfying p 2k−1 + k(αOK ) = "(α)α for (αOK ; −DOK ) = 1; k 2 Z (1.1) (see [R2]). The number of such characters equals the class number h(−D) of K. -
22 Ring Class Fields and the CM Method
18.783 Elliptic Curves Spring 2015 Lecture #22 04/30/2015 22 Ring class fields and the CM method p Let O be an imaginary quadratic order of discriminant D, let K = Q( D), and let L be the splitting field of the Hilbert class polynomial HD(X) over K. In the previous lecture we showed that there is an injective group homomorphism Ψ: Gal(L=K) ,! cl(O) that commutes with the group actions of Gal(L=K) and cl(O) on the set EllO(C) = EllO(L) of roots of HD(X) (the j-invariants of elliptic curves with CM by O). To complete the proof of the the First Main Theorem of Complex Multiplication, which asserts that Ψ is an isomorphism, we need to show that Ψ is surjective; this is equivalent to showing the HD(X) is irreducible over K. At the end of the last lecture we introduced the Artin map p 7! σp, which sends each unramified prime p of K to the unique automorphism σp 2 Gal(L=K) for which Np σp(x) ≡ x mod q; (1) for all x 2 OL and primes q of L dividing pOL (recall that σp is independent of q because Gal(L=K) ,! cl(O) is abelian). Equivalently, σp is the unique element of Gal(L=K) that Np fixes q and induces the Frobenius automorphism x 7! x of Fq := OL=q, which is a generator for Gal(Fq=Fp), where Fp := OK =p. Note that if E=C has CM by O then j(E) 2 L, and this implies that E can be defined 2 3 by a Weierstrass equation y = x + Ax + B with A; B 2 OL.