History of Class Field Theory
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THE INTERSECTION of NORM GROUPS(I) by JAMES AX
THE INTERSECTION OF NORM GROUPS(i) BY JAMES AX 1. Introduction. Let A be a global field (either a finite extension of Q or a field of algebraic functions in one variable over a finite field) or a local field (a local completion of a global field). Let C(A,n) (respectively: A(A, n), JV(A,n) and £(A, n)) be the set of Xe A such that X is the norm of every cyclic (respectively: abelian, normal and arbitrary) extension of A of degree n. We show that (*) C(A, n) = A(A,n) = JV(A,n) = £(A, n) = A" is "almost" true for any global or local field and any natural number n. For example, we prove (*) if A is a number field and %)(n or if A is a function field and n is arbitrary. In the case when (*) is false we are still able to determine C(A, n) precisely. It then turns out that there is a specified X0e A such that C(A,n) = r0/2An u A". Since we always have C(A,n) =>A(A,n) =>N(A,n) r> £(A,n) =>A", there are thus two possibilities for each of the three middle sets. Determining which is true seems to be a delicate question; our results on this problem, which are incomplete, are presented in §5. 2. Preliminaries. We consider an algebraic number field A as a subfield of the field of all complex numbers. If p is a nonarchimedean prime of A then there is a natural injection A -> Ap where Ap denotes the completion of A at p. -
Local Class Field Theory Via Lubin-Tate Theory and Applications
Local Class Field Theory via Lubin-Tate Theory and Applications Misja F.A. Steinmetz Supervisor: Prof. Fred Diamond June 2016 Abstract In this project we will introduce the study of local class field theory via Lubin-Tate theory following Yoshida [Yos08]. We explain how to construct the Artin map for Lubin-Tate extensions and we will show this map gives an isomorphism onto the Weil group of the maximal Lubin-Tate extension of our local field K: We will, furthermore, state (without proof) the other results needed to complete a proof of local class field theory in the classical sense. At the end of the project, we will look at a result from a recent preprint of Demb´el´e,Diamond and Roberts which gives an 1 explicit description of a filtration on H (GK ; Fp(χ)) for K a finite unramified extension of Qp and × χ : GK ! Fp a character. Using local class field theory, we will prove an analogue of this result for K a totally tamely ramified extension of Qp: 1 Contents 1 Introduction 3 2 Local Class Field Theory5 2.1 Formal Groups..........................................5 2.2 Lubin-Tate series.........................................6 2.3 Lubin-Tate Modules.......................................7 2.4 Lubin-Tate Extensions for OK ..................................8 2.5 Lubin-Tate Groups........................................9 2.6 Generalised Lubin-Tate Extensions............................... 10 2.7 The Artin Map.......................................... 11 2.8 Local Class Field Theory.................................... 13 1 3 Applications: Filtration on H (GK ; Fp(χ)) 14 3.1 Definition of the filtration.................................... 14 3.2 Computation of the jumps in the filtration.......................... -
The Kronecker-Weber Theorem
The Kronecker-Weber Theorem Lucas Culler Introduction The Kronecker-Weber theorem is one of the earliest known results in class field theory. It says: Theorem. (Kronecker-Weber-Hilbert) Every abelian extension of the rational numbers Q is con- tained in a cyclotomic extension. Recall that an abelian extension is a finite field extension K/Q such that the galois group Gal(K/Q) th is abelian, and a cyclotomic extension is an extension of the form Q(ζ), where ζ is an n root of unity. This paper consists of two proofs of the Kronecker-Weber theorem. The first is rather involved, but elementary, and uses the theory of higher ramification groups. The second is a simple application of the main results of class field theory, which classifies abelian extension of an arbitrary number field. An Elementary Proof Now we will present an elementary proof of the Kronecker-Weber theoerem, in the spirit of Hilbert’s original proof. The particular strategy used here is given as a series of exercises in Marcus [1]. Minkowski’s Theorem We first prove a classical result due to Minkowski. Theorem. (Minkowski) Any finite extension of Q has nonzero discriminant. In particular, such an extension is ramified at some prime p ∈ Z. Proof. Let K/Q be a finite extension of degree n, and let A = OK be its ring of integers. Consider the embedding: r s A −→ R ⊕ C x 7→ (σ1(x), ..., σr(x), τ1(x), ..., τs(x)) where the σi are the real embeddings of K and the τi are the complex embeddings, with one embedding chosen from each conjugate pair, so that n = r + 2s. -
The Kronecker-Weber Theorem
18.785 Number theory I Fall 2017 Lecture #20 11/15/2017 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) 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], who provided a partial proof for extensions of odd degree. Weber [7] published a proof 1886 that was believed to address the remaining cases; in fact Weber's proof contains some gaps (as noted in [5]), but in any case an alternative proof was given a few years later by Hilbert [1]. The proof we present here is adapted from [6, Ch. 14] 20.1 Local and global Kronecker-Weber theorems We now state the (global) Kronecker-Weber theorem. -
ON R-EXTENSIONS of ALGEBRAIC NUMBER FIELDS Let P Be a Prime
ON r-EXTENSIONS OF ALGEBRAIC NUMBER FIELDS KENKICHI IWASAWA1 Let p be a prime number. We call a Galois extension L of a field K a T-extension when its Galois group is topologically isomorphic with the additive group of £-adic integers. The purpose of the present paper is to study arithmetic properties of such a T-extension L over a finite algebraic number field K. We consider, namely, the maximal unramified abelian ^-extension M over L and study the structure of the Galois group G(M/L) of the extension M/L. Using the result thus obtained for the group G(M/L)> we then define two invariants l(L/K) and m(L/K)} and show that these invariants can be also de termined from a simple formula which gives the exponents of the ^-powers in the class numbers of the intermediate fields of K and L. Thus, giving a relation between the structure of the Galois group of M/L and the class numbers of the subfields of L, our result may be regarded, in a sense, as an analogue, for L, of the well-known theorem in classical class field theory which states that the class number of a finite algebraic number field is equal to the degree of the maximal unramified abelian extension over that field. An outline of the paper is as follows: in §1—§5, we study the struc ture of what we call T-finite modules and find, in particular, invari ants of such modules which are similar to the invariants of finite abelian groups. -
Number Theory
Number Theory Alexander Paulin October 25, 2010 Lecture 1 What is Number Theory Number Theory is one of the oldest and deepest Mathematical disciplines. In the broadest possible sense Number Theory is the study of the arithmetic properties of Z, the integers. Z is the canonical ring. It structure as a group under addition is very simple: it is the infinite cyclic group. The mystery of Z is its structure as a monoid under multiplication and the way these two structure coalesce. As a monoid we can reduce the study of Z to that of understanding prime numbers via the following 2000 year old theorem. Theorem. Every positive integer can be written as a product of prime numbers. Moreover this product is unique up to ordering. This is 2000 year old theorem is the Fundamental Theorem of Arithmetic. In modern language this is the statement that Z is a unique factorization domain (UFD). Another deep fact, due to Euclid, is that there are infinitely many primes. As a monoid therefore Z is fairly easy to understand - the free commutative monoid with countably infinitely many generators cross the cyclic group of order 2. The point is that in isolation addition and multiplication are easy, but together when have vast hidden depth. At this point we are faced with two potential avenues of study: analytic versus algebraic. By analytic I questions like trying to understand the distribution of the primes throughout Z. By algebraic I mean understanding the structure of Z as a monoid and as an abelian group and how they interact. -
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. -
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. -
RECENT THOUGHTS on ABELIAN POINTS 1. Introduction While At
RECENT THOUGHTS ON ABELIAN POINTS 1. Introduction While at MSRI in early 2006, I got asked a very interesting question by Dimitar Jetchev, a Berkeley grad student. It motivated me to study abelian points on al- gebraic curves (and, to a lesser extent, higher-dimensional algebraic varieties), and MSRI's special program on Rational and Integral Points was a convenient setting for this. It did not take me long to ¯nd families of curves without abelian points; I wrote these up in a paper which will appear1 in Math. Research Letters. Since then I have continued to try to put these examples into a larger context. Indeed, I have tried several di®erent larger contexts on for size. When, just a cou- ple of weeks ago, I was completing revisions on the paper, the context of \Kodaira dimension" seemed most worth promoting. Now, after ruminating about my up- coming talk for several days it seems that \Field arithmetic" should also be part of the picture. Needless to say, the ¯nal and optimal context (whatever that might mean!) has not yet been found. So, after having mentally rewritten the beginning of my talk many times, it strikes me that the revisionist approach may not be best: rather, I will for the most part present things in their actual chronological order. 2. Dimitar's Question It was: Question 1. Let C=Q be Selmer's cubic curve: 3X3 + 4Y 3 + 5Z3 = 0: Is there an abelian cubic ¯eld L such that C(L) 6= ;? Or an abelian number ¯eld of any degree? Some basic comments: a ¯nite degree ¯eld extension L=K is abelian if it is Galois with abelian Galois group, i.e., if Aut(L=K) is an abelian group of order [L : K]. -
Modular Forms and the Hilbert Class Field
Modular forms and the Hilbert class field Vladislav Vladilenov Petkov VIGRE 2009, Department of Mathematics University of Chicago Abstract The current article studies the relation between the j−invariant function of elliptic curves with complex multiplication and the Maximal unramified abelian extensions of imaginary quadratic fields related to these curves. In the second section we prove that the j−invariant is a modular form of weight 0 and takes algebraic values at special points in the upper halfplane related to the curves we study. In the third section we use this function to construct the Hilbert class field of an imaginary quadratic number field and we prove that the Ga- lois group of that extension is isomorphic to the Class group of the base field, giving the particular isomorphism, which is closely related to the j−invariant. Finally we give an unexpected application of those results to construct a curious approximation of π. 1 Introduction We say that an elliptic curve E has complex multiplication by an order O of a finite imaginary extension K/Q, if there exists an isomorphism between O and the ring of endomorphisms of E, which we denote by End(E). In such case E has other endomorphisms beside the ordinary ”multiplication by n”- [n], n ∈ Z. Although the theory of modular functions, which we will define in the next section, is related to general elliptic curves over C, throughout the current paper we will be interested solely in elliptic curves with complex multiplication. Further, if E is an elliptic curve over an imaginary field K we would usually assume that E has complex multiplication by the ring of integers in K. -
Algorithms for Ray Class Groups and Hilbert Class Fields 1 Introduction
Algorithms for ray class groups and Hilbert class fields∗ Kirsten Eisentr¨ager† Sean Hallgren‡ Abstract This paper analyzes the complexity of problems from class field theory. Class field theory can be used to show the existence of infinite families of number fields with constant root discriminant. Such families have been proposed for use in lattice-based cryptography and for constructing error-correcting codes. Little is known about the complexity of computing them. We show that computing the ray class group and computing certain subfields of Hilbert class fields efficiently reduce to known computationally difficult problems. These include computing the unit group and class group, the principal ideal problem, factoring, and discrete log. As a consequence, efficient quantum algorithms for these problems exist in constant degree number fields. 1 Introduction The central objects studied in algebraic number theory are number fields, which are finite exten- sions of the rational numbers Q. Class field theory focuses on special field extensions of a given number field K. It can be used to show the existence of infinite families of number fields with constant root discriminant. Such number fields have recently been proposed for applications in cryptography and error correcting codes. In this paper we give algorithms for computing some of the objects required to compute such extensions of number fields. Similar to the approach in computational group theory [BBS09] where there are certain subproblems such as discrete log that are computationally difficult to solve, we identify the subproblems and show that they are the only obstacles. Furthermore, there are quantum algorithms for these subproblems, resulting in efficient quantum algorithms for constant degree number fields for the problems we study.