Algorithmic Enumeration of Ideal Classes for Quaternion Orders
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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. -
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. -
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). -
The Twelfth Problem of Hilbert Reminds Us, Although the Reminder Should
Some Contemporary Problems with Origins in the Jugendtraum Robert P. Langlands The twelfth problem of Hilbert reminds us, although the reminder should be unnecessary, of the blood relationship of three subjects which have since undergone often separate devel• opments. The first of these, the theory of class fields or of abelian extensions of number fields, attained what was pretty much its final form early in this century. The second, the algebraic theory of elliptic curves and, more generally, of abelian varieties, has been for fifty years a topic of research whose vigor and quality shows as yet no sign of abatement. The third, the theory of automorphic functions, has been slower to mature and is still inextricably entangled with the study of abelian varieties, especially of their moduli. Of course at the time of Hilbert these subjects had only begun to set themselves off from the general mathematical landscape as separate theories and at the time of Kronecker existed only as part of the theories of elliptic modular functions and of cyclotomicfields. It is in a letter from Kronecker to Dedekind of 1880,1 in which he explains his work on the relation between abelian extensions of imaginary quadratic fields and elliptic curves with complex multiplication, that the word Jugendtraum appears. Because these subjects were so interwoven it seems to have been impossible to disentangle the different kinds of mathematics which were involved in the Jugendtraum, especially to separate the algebraic aspects from the analytic or number theoretic. Hilbert in particular may have been led to mistake an accident, or perhaps necessity, of historical development for an “innigste gegenseitige Ber¨uhrung.” We may be able to judge this better if we attempt to view the mathematical content of the Jugendtraum with the eyes of a sophisticated contemporary mathematician. -
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. -
Class Field Theory and the Local-Global Property
Class Field Theory and the Local-Global Property Noah Taylor Discussed with Professor Frank Calegari March 18, 2017 1 Classical Class Field Theory 1.1 Lubin Tate Theory In this section we derive a concrete description of a relationship between the Galois extensions of a local field K with abelian Galois groups and the open subgroups of K×. We assume that K is a finite extension of Qp, OK has maximal ideal m, and we suppose the residue field k of K has size q, a power of p. Lemma 1.1 (Hensel). If f(x) is a polynomial in O [x] and a 2 O with f(a) < 1 then there is K K f 0(a)2 some a with f(a ) = 0 and ja − a j ≤ f(a) . 1 1 1 f 0(a) f(a) Proof. As in Newton's method, we continually replace a with a− f 0(a) . Careful consideration of the valuation of f(a) shows that it decreases while f 0(a) does not, so that f(a) tends to 0. Because K is complete, the sequence of a's we get has a limit a1, and we find that f(a1) = 0. The statement about the distance between a and a1 follows immediately. We can apply this first to the polynomial f(x) = xq − x and a being any representative of any element in k to find that K contains the q − 1'st roots of unity. It also follows that if L=K is an unramified extension of degree n, then L = K(ζ) for a qn − 1'st root of unity ζ. -
Course Notes, Global Class Field Theory Caltech, Spring 2015/16
COURSE NOTES, GLOBAL CLASS FIELD THEORY CALTECH, SPRING 2015/16 M. FLACH 1. Global class field theory in classical terms Recall the isomorphism Z Z × −!∼ Q Q 7! 7! a ( =m ) Gal( (ζm)= ); a (ζm ζm) where ζm is a primitive m-th root of unity. This is the reciprocity isomorphism of global class field theory for the number field Q. What could a generalization of this to any number field K be? Of course one has a natural map (going the other way) ∼ × Gal(K(ζm)=K) ! Aut(µm) = (Z=mZ) ; σ 7! (ζ 7! σ(ζ)) called the cyclotomic character. It is even an isomorphism if K is linearly disjoint from Q(ζm). But that's not class field theory, or rather it's only a tiny part of class field theory for the field K. The proper, and much more stunning generalisation is a reciprocity isomorphism ∼ Hm −! Gal(K(m)=K) between so called ray class groups Hm of modulus m and Galois groups of corre- sponding finite abelian extensions K(m)=K. In a first approximation one can think of a modulus m as an integral ideal in OK but it is slightly more general. The ray m × class group H isn't just (OK =m) although the two groups are somewhat related. In any case Hm is explicit, or one should say easily computable as soon as one can O× O compute the unit group K and the class group Cl( K ) of the field K. Note that both are trivial for K = Q, so that ray class groups are particularly easy for K = Q. -
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. -
Brumer–Stark Units and Hilbert's 12Th Problem
Brumer{Stark Units and Hilbert's 12th Problem Samit Dasgupta Mahesh Kakde March 3, 2021 Abstract Let F be a totally real field of degree n and p an odd prime. We prove the p- part of the integral Gross{Stark conjecture for the Brumer{Stark p-units living in CM abelian extensions of F . In previous work, the first author showed that such a result implies an exact p-adic analytic formula for these Brumer{Stark units up to a bounded root of unity error, including a \real multiplication" analogue of Shimura's celebrated reciprocity law in the theory of Complex Multiplication. In this paper we show that the Brumer{Stark units, along with n−1 other easily described elements (these are simply square roots of certain elements of F ) generate the maximal abelian extension of F . We therefore obtain an unconditional solution to Hilbert's 12th problem for totally real fields, albeit one that involves p-adic integration, for infinitely many primes p. Our method of proof of the integral Gross{Stark conjecture is a generalization of our previous work on the Brumer{Stark conjecture. We apply Ribet's method in the context of group ring valued Hilbert modular forms. A key new construction here is the definition of a Galois module rL that incorporates an integral version of the Greenberg{Stevens L -invariant into the theory of Ritter{Weiss modules. This allows for the reinterpretation of Gross's conjecture as the vanishing of the Fitting ideal of rL . This vanishing is obtained by constructing a quotient of rL whose Fitting ideal vanishes using the Galois representations associated to cuspidal Hilbert modular forms.