Efforts in the Direction of Hilbert's Tenth Problem
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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. -
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. -
Introduction. There Are at Least Three Different Problems with Which One Is Confronted in the Study of L-Functions: the Analytic
L-Functions and Automorphic Representations∗ R. P. Langlands Introduction. There are at least three different problems with which one is confronted in the study of L•functions: the analytic continuation and functional equation; the location of the zeroes; and in some cases, the determination of the values at special points. The first may be the easiest. It is certainly the only one with which I have been closely involved. There are two kinds of L•functions, and they will be described below: motivic L•functions which generalize the Artin L•functions and are defined purely arithmetically, and automorphic L•functions, defined by data which are largely transcendental. Within the automorphic L• functions a special class can be singled out, the class of standard L•functions, which generalize the Hecke L•functions and for which the analytic continuation and functional equation can be proved directly. For the other L•functions the analytic continuation is not so easily effected. However all evidence indicates that there are fewer L•functions than the definitions suggest, and that every L•function, motivic or automorphic, is equal to a standard L•function. Such equalities are often deep, and are called reciprocity laws, for historical reasons. Once a reciprocity law can be proved for an L•function, analytic continuation follows, and so, for those who believe in the validity of the reciprocity laws, they and not analytic continuation are the focus of attention, but very few such laws have been established. The automorphic L•functions are defined representation•theoretically, and it should be no surprise that harmonic analysis can be applied to some effect in the study of reciprocity laws. -
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. -
22 Ring Class Fields and the CM Method
18.783 Elliptic Curves Spring 2017 Lecture #22 05/03/2017 22 Ring class fields and the CM method Let O be an imaginary quadratic order of discriminant D, and let EllO(C) := fj(E) 2 C : End(E) = Cg. In the previous lecture we proved that the Hilbert class polynomial Y HD(X) := HO(X) := X − j(E) j(E)2EllO(C) has integerp coefficients. We then defined L to be the splitting field of HD(X) over the field K = Q( D), and 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). 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, equivalently, that 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 (prime ideal of OK ) to the corresponding Frobenius element σp, which is the unique element of Gal(L=K) for which Np σp(x) ≡ x mod q; (1) for all x 2 OL and primes qjp (prime ideals of OL that divide the ideal pOL); the existence of a single σp 2 Gal(L=K) satisfying (1) for all qjp follows from the fact that Gal(L=K) ,! cl(O) is abelian. The Frobenius element σp can also be characterized as follows: for each prime qjp the finite field Fq := OL=q is an extension of the finite field Fp := OK =p and the automorphism σ¯p 2 Gal(Fq=Fp) defined by σ¯p(¯x) = σ(x) (where x 7! x¯ is the reduction Np map OL !OL=q), is the Frobenius automorphism x 7! x generating Gal(Fq=Fp). -
L-Functions and Non-Abelian Class Field Theory, from Artin to Langlands
L-functions and non-abelian class field theory, from Artin to Langlands James W. Cogdell∗ Introduction Emil Artin spent the first 15 years of his career in Hamburg. Andr´eWeil charac- terized this period of Artin's career as a \love affair with the zeta function" [77]. Claude Chevalley, in his obituary of Artin [14], pointed out that Artin's use of zeta functions was to discover exact algebraic facts as opposed to estimates or approxi- mate evaluations. In particular, it seems clear to me that during this period Artin was quite interested in using the Artin L-functions as a tool for finding a non- abelian class field theory, expressed as the desire to extend results from relative abelian extensions to general extensions of number fields. Artin introduced his L-functions attached to characters of the Galois group in 1923 in hopes of developing a non-abelian class field theory. Instead, through them he was led to formulate and prove the Artin Reciprocity Law - the crowning achievement of abelian class field theory. But Artin never lost interest in pursuing a non-abelian class field theory. At the Princeton University Bicentennial Conference on the Problems of Mathematics held in 1946 \Artin stated that `My own belief is that we know it already, though no one will believe me { that whatever can be said about non-Abelian class field theory follows from what we know now, since it depends on the behavior of the broad field over the intermediate fields { and there are sufficiently many Abelian cases.' The critical thing is learning how to pass from a prime in an intermediate field to a prime in a large field. -
1. Basic Ramification Theory in This Section L/K Is a Nite Galois Extension of Number Elds of Degree N
LECTURE 4 YIHANG ZHU 1. Basic ramification theory In this section L=K is a nite Galois extension of number elds of degree n. The Galois group Gal(L=K) acts on various invariants of L, for instance the group of fractional ideals IL and the class group Cl(L). If P is a prime of L above p of K, then any element of Gal(L=K) sends P to another prime above p. We have Proposition 1.1. Let L=K be a nite Galois extension of number elds. Then for any prime of , the Galois group acts transitively on the set p K Gal(L=K) fPig1≤i≤g of primes of L above p. In particular, the inertia degrees fi are the same, denoted by f = f(p; L=K), and by unique factorization, the ramication degrees ei are the same, denoted by e = e(p; L=K). The fundamental identity reduces to efg = n: Let P be a prime of L above a prime p of K. Let e; f; g be as above. Denition 1.2. The stabilizer of P in Gal(L=K) is called the decomposition group of B, denoted by D(P). The corresponding subeld LD(P) of L is called the decomposition eld, denoted by ZP. −1 Remark 1.3. For σ 2 Gal(L=K), D(σP) = σD(P)σ and ZσP = σ(ZP): The group P is the stabilizer in a group of order n on an orbit of cardinality g, so its order is n=g = ef. -
The Chebotarëv Density Theorem Applications
UNIVERSITA` DEGLI STUDI ROMA TRE FACOLTA` DI SCIENZE MATEMATICHE FISICHE NATURALI Graduation Thesis in Mathematics by Alfonso Pesiri The Chebotar¨evDensity Theorem Applications Supervisor Prof. Francesco Pappalardi The Candidate The Supervisor ACADEMIC YEAR 2006 - 2007 OCTOBER 2007 AMS Classification: primary 11R44; 12F10; secondary 11R45; 12F12. Key Words: Chebotar¨evDensity Theorem, Galois theory, Transitive groups. Contents Introduction . 1 1 Algebraic background 10 1.1 The Frobenius Map . 10 1.2 The Artin Symbol in Abelian Extensions . 12 1.3 Quadratic Reciprocity . 17 1.4 Cyclotomic Extensions . 20 1.5 Dedekind Domains . 22 1.6 The Frobenius Element . 26 2 Chebotar¨ev’s Density Theorem 32 2.1 Symmetric Polynomials . 32 2.2 Dedekind’s Theorem . 33 2.3 Frobenius’s Theorem . 36 2.4 Chebotar¨ev’s Theorem . 39 2.5 Frobenius and Chebotar¨ev . 40 2.6 Dirichlet’s Theorem on Primes in Arithmetic Progression . 41 2.7 Hint of the Proof . 46 3 Applications 48 3.1 Charming Consequences . 48 3.2 Primes and Quadratic Forms . 52 3.3 A Probabilistic Approach . 55 3.4 Transitive Groups . 61 1 4 Inverse Galois Problem 71 4.1 Computing Galois Groups . 71 4.2 Groups of Prime Degree Polynomials . 83 A Roots on Finite Fields 88 B Galois Groups on Finite Fields 90 C The Chebotar¨evTest in Maple 92 Bibliography 113 2 Introduction The problem of solving polynomial equations has interested mathemati- cians for ages. The Babylonians had methods for solving some quadratic equations in 1600 BC. The ancient Greeks had other methods for solving quadratic equations and their geometric approach also gave them a tool for solving some cubic equations. -
On the Genesis of Robert P. Langlands' Conjectures and His Letter to André
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY http://dx.doi.org/10.1090/bull/1609 Article electronically published on January 25, 2018 ON THE GENESIS OF ROBERT P. LANGLANDS’ CONJECTURES AND HIS LETTER TO ANDREWEIL´ JULIA MUELLER Abstract. This article is an introduction to the early life and work of Robert P. Langlands, creator and founder of the Langlands program. The story is, to a large extent, told by Langlands himself, in his own words. Our focus is on two of Langlands’ major discoveries: automorphic L-functions and the principle of functoriality. It was Langlands’ desire to communicate his excitement about his newly discovered objects that resulted in his famous letter to Andr´e Weil. This article is aimed at a general mathematical audience and we have purposely not included the more technical aspects of Langlands’ work. Contents 1. Introduction 1 2. Langlands’ early years (1936–1960) 3 3. Overview 11 Acknowledgments 36 About the author 36 References 36 1. Introduction This article is about the life and work of Robert Langlands, covering the first 30 years of his life from 1936 to 1966 and his work from 1960 to 1966. His letter to Andr´e Weil dates to January 1967 and the Langlands program was launched soon afterward. Section 2 of this article focuses on Langlands’ early years, from 1936 to 1960, and the story is told by Langlands himself. The material is taken from an interview given by Langlands to a student, Farzin Barekat, at the University of British Columbia (UBC), Langlands’ alma mater, in the early 2000s. -
Hilbert Class Fields of Imaginary Quadratic Fields and Reflex Fields of Certain Sextic CM Fields
University of Vermont ScholarWorks @ UVM Graduate College Dissertations and Theses Dissertations and Theses 2017 Hilbert Class Fields of Imaginary Quadratic Fields and Reflex ieldF s of Certain Sextic CM Fields Garvin Gaston University of Vermont Follow this and additional works at: https://scholarworks.uvm.edu/graddis Part of the Mathematics Commons Recommended Citation Gaston, Garvin, "Hilbert Class Fields of Imaginary Quadratic Fields and Reflex ieF lds of Certain Sextic CM Fields" (2017). Graduate College Dissertations and Theses. 808. https://scholarworks.uvm.edu/graddis/808 This Thesis is brought to you for free and open access by the Dissertations and Theses at ScholarWorks @ UVM. It has been accepted for inclusion in Graduate College Dissertations and Theses by an authorized administrator of ScholarWorks @ UVM. For more information, please contact [email protected]. Hilbert Class Fields of Imaginary Quadratic Fields and Reflex Fields of Certain Sextic CM Fields A Thesis Presented by Garvin Gaston to The Faculty of the Graduate College of The University of Vermont In Partial Fulfillment of the Requirements for the Degree of Master of Science Specializing in Mathematics October, 2017 Defense Date: August 2, 2017 Thesis Examination Committee: Christelle Vincent, Ph.D., Advisor Christian Skalka, Ph.D., Chairperson Richard Foote, Ph.D. Cynthia J. Forehand, Ph.D., Dean of Graduate College Abstract In this thesis we look at particular details of class field theory for complex multipli- cation fields. We begin by giving some background on fields, abelian varieties, and complex multiplication. We then turn to the first task of this thesis and give an implementation in Sage of a classical algorithm to compute the Hilbert class field of a quadratic complex multiplication field using the j-invariant of elliptic curves with complex multiplication by the ring of integers of the field, and we include three explicit examples to illustrate the algorithm. -
A RECIPROCITY LAW for CERTAIN FROBENIUS EXTENSIONS 1. Introduction for Any Number Field F, Let FA Be Its Adele Ring. Let E/F Be
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 6, June 1996 ARECIPROCITYLAW FOR CERTAIN FROBENIUS EXTENSIONS YUANLI ZHANG (Communicated by Dennis A. Hejhal) Abstract. Let E/F be a finite Galois extension of algebraic number fields with Galois group G. Assume that G is a Frobenius group and H is a Frobenius complement of G.LetF(H) be the maximal normal nilpotent subgroup of H. If H/F (H) is nilpotent, then every Artin L-function attached to an irreducible representation of G arises from an automorphic representation over F , i.e., the Langlands’ reciprocity conjecture is true for such Galois extensions. 1. Introduction For any number field F ,letFA be its adele ring. Let E/F be a finite Galois extension of algebraic number fields, G =Gal(E/F), and Irr(G)bethesetof isomorphism classes of irreducible complex representations of G. In the early 1900’s, E. Artin associated an L-function L(s, ρ, E/F ), which is defined originally by an Euler product of the local L-functions over all nonarchimedean places of F ,toevery ρ Irr(G). These L-functions play a fundamental role in describing the distribution of∈ prime ideal factorizations in such an extension, just as the classical L-functions of Dirichlet determine the distribution of primes in a given arithmetic progression and the prime ideal factorizations in cyclotomic fields. However, unlike the L-functions of Dirichlet, we do not know in general if the Artin L-functions extend to entire functions. In fact, we have the important: Artin’s conjecture. For any irreducible complex linear representation ρ of G, L(s, ρ, E/F ) extends to an entire function whenever ρ =1.