Brumer–Stark Units and Hilbert's 12Th Problem
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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. -
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. -
Arxiv:1501.01388V2 [Math.NT] 25 Jul 2016 E Od N Phrases
On the Iwasawa theory of CM fields for supersingular primes KÂZIM BÜYÜKBODUK Abstract. The goal of this article is two-fold: First, to prove a (two-variable) main conjecture for a CM field F without assuming the p-ordinary hypothesis of Katz, making use of what we call the Rubin-Stark -restricted Kolyvagin systems which is constructed out of the conjectural Rubin-Stark EulerL system of rank g. (We are also able to obtain weaker unconditional results in this direction.) Second objective is to prove the Park- Shahabi plus/minus main conjecture for a CM elliptic curve E defined over a general totally real field again using (a twist of the) Rubin-Stark Kolyvagin system. This latter result has consequences towards the Birch and Swinnerton-Dyer conjecture for E. Contents 1. Introduction 2 Notation 3 Statements of the results 3 Overview of the methods and layout of the paper 5 Acknowledgements 5 1.1. Notation and Hypotheses 6 2. Selmer structures and comparing Selmer groups 7 2.1. Structure of the semi-local cohomology groups 7 2.2. Modified Selmer structures for Gm 8 cyc 2.3. Modified Selmer structures for Gm along F and F∞ 9 2.4. Modified Selmer structures for E 12 2.5. Global duality and comparison of Selmer groups 14 3. Rubin-Stark Euler system of rank r 16 3.1. Strong Rubin-Stark Conjectures 17 4. Kolyvagin systems for Gm and E 19 4.1. Rubin-Stark -restricted Kolyvagin systems 22 L arXiv:1501.01388v2 [math.NT] 25 Jul 2016 5. Gras’ conjecture and CM main conjectures over F 24 5.1. -
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. -
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. -
Arxiv:2010.00657V2 [Math.NT] 6 Nov 2020 on the Brumer–Stark
On the Brumer–Stark Conjecture Samit Dasgupta Mahesh Kakde November 9, 2020 Abstract Let H/F be a finite abelian extension of number fields with F totally real and H a CM field. Let S and T be disjoint finite sets of places of F satisfying the standard H/F conditions. The Brumer–Stark conjecture states that the Stickelberger element ΘS,T annihilates the T -smoothed class group ClT (H). We prove this conjecture away from p = 2, that is, after tensoring with Z[1/2]. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting ideal of the minus part of the Pontryagin dual of ClT (H) ⊗ Z[1/2] in terms of Stickelberger elements. We also show that this stronger result implies Rubin’s higher rank version of the Brumer–Stark conjecture, again away from 2. Our technique is a generalization of Ribet’s method, building upon on our earlier work on the Gross–Stark conjecture. Here we work with group ring valued Hilbert modular forms as introduced by Wiles. A key aspect of our approach is the construction of congruences between cusp forms and Eisenstein series that are stronger than usually expected, arising as shadows of the trivial zeroes of p-adic L-functions. These stronger congruences are essential to proving that the cohomology classes we construct are unramified at p. Contents arXiv:2010.00657v2 [math.NT] 6 Nov 2020 1 Introduction 3 1.1 MainResult.................................... 6 1.2 TheRubin–StarkConjecture. 8 1.3 SummaryofProof ................................ 9 1.4 Acknowledgements ............................... -
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]. -
Lectures on Arithmetic Noncommutative Geometry Matilde Marcolli
Lectures on Arithmetic Noncommutative Geometry Matilde Marcolli And indeed there will be time To wonder \Do I dare?" and, \Do I dare?" Time to turn back and descend the stair. ... Do I dare Disturb the Universe? ... For I have known them all already, known them all; Have known the evenings, mornings, afternoons, I have measured out my life with coffee spoons. ... I should have been a pair of ragged claws Scuttling across the floors of silent seas. ... No! I am not Prince Hamlet, nor was meant to be; Am an attendant lord, one that will do To swell a progress, start a scene or two ... At times, indeed, almost ridiculous{ Almost, at times, the Fool. ... We have lingered in the chambers of the sea By sea-girls wreathed with seaweed red and brown Till human voices wake us, and we drown. (T.S. Eliot, \The Love Song of J. Alfred Prufrock") Contents Chapter 1. Ouverture 5 1. The NCG dictionary 7 2. Noncommutative spaces 8 3. Spectral triples 9 Chapter 2. Noncommutative modular curves 17 1. Modular curves 17 2. The noncommutative boundary of modular curves 24 3. Modular interpretation: noncommutative elliptic curves 24 4. Limiting modular symbols 29 5. Hecke eigenforms 40 6. Selberg zeta function 43 7. The modular complex and K-theory of C∗-algebras 44 8. Intermezzo: Chaotic Cosmology 45 Chapter 3. Quantum statistical mechanics and Galois theory 53 1. Quantum Statistical Mechanics 54 2. The Bost{Connes system 58 3. Noncommutative Geometry and Hilbert's 12th problem 63 4. The GL2 system 65 5. Quadratic fields 72 Chapter 4. -
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).