An Introduction to the Theory of Valued Fields
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OSTROWSKI for NUMBER FIELDS 1. Introduction in 1916, Ostrowski [6] Classified the Nontrivial Absolute Values on Q: up to Equival
OSTROWSKI FOR NUMBER FIELDS KEITH CONRAD 1. Introduction In 1916, Ostrowski [6] classified the nontrivial absolute values on Q: up to equivalence, they are the usual (archimedean) absolute value and the p-adic absolute values for different primes p, with none of these being equivalent to each other. We will see how this theorem extends to a number field K, giving a list of all the nontrivial absolute values on K up to equivalence: for each nonzero prime ideal p in OK there is a p-adic absolute value, real embeddings of K and complex embeddings of K up to conjugation lead to archimedean absolute values on K, and every nontrivial absolute value on K is equivalent to a p-adic, real, or complex absolute value. 2. Defining nontrivial absolute values on K For each nonzero prime ideal p in OK , a p-adic absolute value on K is defined in terms × of a p-adic valuation ordp that is first defined on OK − f0g and extended to K by taking ratios. m Definition 2.1. For x 2 OK − f0g, define ordp(x) := m where xOK = p a with m ≥ 0 and p - a. We have ordp(xy) = ordp(x)+ordp(y) for nonzero x and y in OK by unique factorization of × ideals in OK . This lets us extend ordp to K by using ratios of nonzero numbers in OK : for × α 2 K , write α = x=y for nonzero x and y in OK and set ordp(α) := ordp(x) − ordp(y). To 0 0 0 0 0 0 see this is well-defined, if x=y = x =y for nonzero x; y; x , and y in OK then xy = x y 0 0 in OK , which implies ordp(x) + ordp(y ) = ordp(x ) + ordp(y), so ordp(x) − ordp(y) = 0 0 × × ordp(x ) − ordp(y ). -
Class Numbers of CM Algebraic Tori, CM Abelian Varieties and Components of Unitary Shimura Varieties
CLASS NUMBERS OF CM ALGEBRAIC TORI, CM ABELIAN VARIETIES AND COMPONENTS OF UNITARY SHIMURA VARIETIES JIA-WEI GUO, NAI-HENG SHEU, AND CHIA-FU YU Abstract. We give a formula for the class number of an arbitrary CM algebraic torus over Q. This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result. 1. Introduction An algebraic torus T over a number field k is a connected linear algebraic group over k such d that T k k¯ isomorphic to (Gm) k k¯ over the algebraic closure k¯ of k for some integer d 1. The class⊗ number, h(T ), of T is by⊗ definition, the cardinality of T (k) T (A )/U , where A≥ \ k,f T k,f is the finite adele ring of k and UT is the maximal open compact subgroup of T (Ak,f ). Asa natural generalization for the class number of a number field, Takashi Ono [18, 19] studied the class numbers of algebraic tori. Let K/k be a finite extension and let RK/k denote the Weil restriction of scalars form K to k, then we have the following exact sequence of tori defined over k 1 R(1) (G ) R (G ) G 1, −→ K/k m,K −→ K/k m,K −→ m,k −→ where R(1) (G ) is the kernel of the norm map N : R (G ) G . -
1 Absolute Values and Valuations
Algebra in positive Characteristic Fr¨uhjahrssemester 08 Lecturer: Prof. R. Pink Organizer: P. Hubschmid Absolute values, valuations and completion F. Crivelli ([email protected]) April 21, 2008 Introduction During this talk I’ll introduce the basic definitions and some results about valuations, absolute values of fields and completions. These notions will give two basic examples: the rational function field Fq(T ) and the field Qp of the p-adic numbers. 1 Absolute values and valuations All along this section, K denote a field. 1.1 Absolute values We begin with a well-known Definition 1. An absolute value of K is a function | | : K → R satisfying these properties, ∀ x, y ∈ K: 1. |x| = 0 ⇔ x = 0, 2. |x| ≥ 0, 3. |xy| = |x| |y|, 4. |x + y| ≤ |x| + |y|. (Triangle inequality) Note that if we set |x| = 1 for all 0 6= x ∈ K and |0| = 0, we have an absolute value on K, called the trivial absolute value. From now on, when we speak about an absolute value | |, we assume that | | is non-trivial. Moreover, if we define d : K × K → R by d(x, y) = |x − y|, x, y ∈ K, d is a metric on K and we have a topological structure on K. We have also some basic properties that we can deduce directly from the definition of an absolute value. 1 Lemma 1. Let | | be an absolute value on K. We have 1. |1| = 1, 2. |ζ| = 1, for all ζ ∈ K with ζd = 1 for some 0 6= d ∈ N (ζ is a root of unity), 3. -
4.3 Absolute Value Equations 373
Section 4.3 Absolute Value Equations 373 4.3 Absolute Value Equations In the previous section, we defined −x, if x < 0. |x| = (1) x, if x ≥ 0, and we saw that the graph of the absolute value function defined by f(x) = |x| has the “V-shape” shown in Figure 1. y 10 x 10 Figure 1. The graph of the absolute value function f(x) = |x|. It is important to note that the equation of the left-hand branch of the “V” is y = −x. Typical points on this branch are (−1, 1), (−2, 2), (−3, 3), etc. It is equally important to note that the right-hand branch of the “V” has equation y = x. Typical points on this branch are (1, 1), (2, 2), (3, 3), etc. Solving |x| = a We will now discuss the solutions of the equation |x| = a. There are three distinct cases to discuss, each of which depends upon the value and sign of the number a. • Case I: a < 0 If a < 0, then the graph of y = a is a horizontal line that lies strictly below the x-axis, as shown in Figure 2(a). In this case, the equation |x| = a has no solutions because the graphs of y = a and y = |x| do not intersect. 1 Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ Version: Fall 2007 374 Chapter 4 Absolute Value Functions • Case II: a = 0 If a = 0, then the graph of y = 0 is a horizontal line that coincides with the x-axis, as shown in Figure 2(b). -
Absolute Values and Discrete Valuations
18.785 Number theory I Fall 2015 Lecture #1 09/10/2015 1 Absolute values and discrete valuations 1.1 Introduction At its core, number theory is the study of the ring Z and its fraction field Q. Many questions about Z and Q naturally lead one to consider finite extensions K=Q, known as number fields, and their rings of integers OK , which are finitely generated extensions of Z (the integral closure of Z in K, as we shall see). There is an analogous relationship between the ring Fq[t] of polynomials over a finite field Fq, its fraction field Fq(t), and the finite extensions of Fq(t), which are known as (global) function fields; these fields play an important role in arithmetic geometry, as they are are categorically equivalent to nice (smooth, projective, geometrically integral) curves over finite fields. Number fields and function fields are collectively known as global fields. Associated to each global field k is an infinite collection of local fields corresponding to the completions of k with respect to its absolute values; for the field of rational numbers Q, these are the field of real numbers R and the p-adic fields Qp (as you will prove on Problem Set 1). The ring Z is a principal ideal domain (PID), as is Fq[t]. These rings have dimension one, which means that every nonzero prime ideal is maximal; thus each nonzero prime ideal has an associated residue field, and for both Z and Fq[t] these residue fields are finite. In the case of Z we have residue fields Z=pZ ' Fp for each prime p, and for Fq[t] we have residue fields Fqd associated to each irreducible polynomial of degree d. -
An Introduction to the P-Adic Numbers Charles I
Union College Union | Digital Works Honors Theses Student Work 6-2011 An Introduction to the p-adic Numbers Charles I. Harrington Union College - Schenectady, NY Follow this and additional works at: https://digitalworks.union.edu/theses Part of the Logic and Foundations of Mathematics Commons Recommended Citation Harrington, Charles I., "An Introduction to the p-adic Numbers" (2011). Honors Theses. 992. https://digitalworks.union.edu/theses/992 This Open Access is brought to you for free and open access by the Student Work at Union | Digital Works. It has been accepted for inclusion in Honors Theses by an authorized administrator of Union | Digital Works. For more information, please contact [email protected]. AN INTRODUCTION TO THE p-adic NUMBERS By Charles Irving Harrington ********* Submitted in partial fulllment of the requirements for Honors in the Department of Mathematics UNION COLLEGE June, 2011 i Abstract HARRINGTON, CHARLES An Introduction to the p-adic Numbers. Department of Mathematics, June 2011. ADVISOR: DR. KARL ZIMMERMANN One way to construct the real numbers involves creating equivalence classes of Cauchy sequences of rational numbers with respect to the usual absolute value. But, with a dierent absolute value we construct a completely dierent set of numbers called the p-adic numbers, and denoted Qp. First, we take p an intuitive approach to discussing Qp by building the p-adic version of 7. Then, we take a more rigorous approach and introduce this unusual p-adic absolute value, j jp, on the rationals to the lay the foundations for rigor in Qp. Before starting the construction of Qp, we arrive at the surprising result that all triangles are isosceles under j jp. -
Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves Athanasios Angelakis
Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves Athanasios Angelakis To cite this version: Athanasios Angelakis. Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves. Group Theory [math.GR]. Université de Bordeaux; Universiteit Leiden (Leyde, Pays-Bas), 2015. English. NNT : 2015BORD0180. tel-01359692 HAL Id: tel-01359692 https://tel.archives-ouvertes.fr/tel-01359692 Submitted on 23 Sep 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves Proefschrift ter verkrijging van de graad van Doctor aan de Universiteit Leiden op gezag van Rector Magnificus prof. mr. C.J.J.M. Stolker, volgens besluit van het College voor Promoties te verdedigen op woensdag 2 september 2015 klokke 15:00 uur door Athanasios Angelakis geboren te Athene in 1979 Samenstelling van de promotiecommissie: Promotor: Prof. dr. Peter Stevenhagen (Universiteit Leiden) Promotor: Prof. dr. Karim Belabas (Universit´eBordeaux I) Overige leden: Prof. -
MAT 240 - Algebra I Fields Definition
MAT 240 - Algebra I Fields Definition. A field is a set F , containing at least two elements, on which two operations + and · (called addition and multiplication, respectively) are defined so that for each pair of elements x, y in F there are unique elements x + y and x · y (often written xy) in F for which the following conditions hold for all elements x, y, z in F : (i) x + y = y + x (commutativity of addition) (ii) (x + y) + z = x + (y + z) (associativity of addition) (iii) There is an element 0 ∈ F , called zero, such that x+0 = x. (existence of an additive identity) (iv) For each x, there is an element −x ∈ F such that x+(−x) = 0. (existence of additive inverses) (v) xy = yx (commutativity of multiplication) (vi) (x · y) · z = x · (y · z) (associativity of multiplication) (vii) (x + y) · z = x · z + y · z and x · (y + z) = x · y + x · z (distributivity) (viii) There is an element 1 ∈ F , such that 1 6= 0 and x·1 = x. (existence of a multiplicative identity) (ix) If x 6= 0, then there is an element x−1 ∈ F such that x · x−1 = 1. (existence of multiplicative inverses) Remark: The axioms (F1)–(F-5) listed in the appendix to Friedberg, Insel and Spence are the same as those above, but are listed in a different way. Axiom (F1) is (i) and (v), (F2) is (ii) and (vi), (F3) is (iii) and (vii), (F4) is (iv) and (ix), and (F5) is (vii). Proposition. Let F be a field. -
LECTURES on HEIGHT ZETA FUNCTIONS of TORIC VARIETIES by Yuri Tschinkel
S´eminaires & Congr`es 6, 2002, p. 227–247 LECTURES ON HEIGHT ZETA FUNCTIONS OF TORIC VARIETIES by Yuri Tschinkel Abstract.— We explain the main ideas and techniques involved in recent proofs of asymptotics of rational points of bounded height on toric varieties. 1. Introduction Toric varieties are an ideal testing ground for conjectures: their theory is sufficiently rich to reflect general phenomena and sufficiently rigid to allow explicit combinato- rial computations. In these notes I explain a conjecture in arithmetic geometry and describe its proof for toric varieties. Acknowledgments. — Iwould like to thank the organizers of the Summer School for the invitation. The results concerning toric varieties were obtained in collaboration with V. Batyrev. It has been a great pleasure and privilege to work with A. Chambert- Loir, B. Hassett and M. Strauch — Iam very much indebted to them. My research was partially supported by the NSA. 1.1. Counting problems Example 1.1.1.—LetX ⊂ Pn be a smooth hypersurface given as the zero set of a homogeneous form f of degree d (with coefficients in Z). Let N(X, B)=#{x | f(x)=0, max(|xj |) B} n+1 (where x =(x0,...,xn) ∈ Z /(±1) with gcd(xj) = 1) be the number of Q-rational points on X of “height” B. Heuristically, the probability that f represents 0 is about B−d and the number of “events” about Bn+1. Thus we expect that lim N(X, B) ∼ Bn+1−d. B→∞ 2000 Mathematics Subject Classification.—14G05, 11D45, 14M25, 11D57. Key words and phrases.—Rational points, heights, toric varieties, zeta functions. -
ADELIC ANALYSIS and FUNCTIONAL ANALYSIS on the FINITE ADELE RING Ilwoo
Opuscula Math. 38, no. 2 (2018), 139–185 https://doi.org/10.7494/OpMath.2018.38.2.139 Opuscula Mathematica ADELIC ANALYSIS AND FUNCTIONAL ANALYSIS ON THE FINITE ADELE RING Ilwoo Cho Communicated by P.A. Cojuhari Abstract. In this paper, we study operator theory on the -algebra , consisting of all ∗ MP measurable functions on the finite Adele ring AQ, in extended free-probabilistic sense. Even though our -algebra is commutative, our Adelic-analytic data and properties on ∗ MP MP are understood as certain free-probabilistic results under enlarged sense of (noncommutative) free probability theory (well-covering commutative cases). From our free-probabilistic model on AQ, we construct the suitable Hilbert-space representation, and study a C∗-algebra M generated by under representation. In particular, we focus on operator-theoretic P MP properties of certain generating operators on M . P Keywords: representations, C∗-algebras, p-adic number fields, the Adele ring, the finite Adele ring. Mathematics Subject Classification: 05E15, 11G15, 11R47, 11R56, 46L10, 46L54, 47L30, 47L55. 1. INTRODUCTION The main purposes of this paper are: (i) to construct a free-probability model (under extended sense) of the -algebra ∗ consisting of all measurable functions on the finite Adele ring A , implying MP Q number-theoretic information from the Adelic analysis on AQ, (ii) to establish a suitable Hilbert-space representation of , reflecting our MP free-distributional data from (i) on , MP (iii) to construct-and-study a C∗-algebra M generated by under our representa- P MP tion of (ii), and (iv) to consider free distributions of the generating operators of M of (iii). -
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. -
Math 676. a Lattice in the Adele Ring Let K Be a Number Field, and Let S Be a Finite Set of Places of K Containing the Set
Math 676. A lattice in the adele ring Let K be a number field, and let S be a finite set of places of K containing the set S of archimedean 1 places. The S-adele ring AK;S = v S Kv v S Ov is an open subring of the adele ring AK , and it meets Q 2 × Q 62 the additive subgroup K in the subring OK;S of S-integers. In class we saw that K is discrete in AK and OK;S is discrete in AK;S . The purpose of this handout is to explain why each of these discrete subgroups is co-compact. The main input will be that OK;S is a lattice in v S Kv, a fact that we saw in an earlier handout (and that was a key ingredient in the proof of the S-unitQ theorem).2 1. The case of S-adeles We first study AK;S =OK;S . This is a locally compact Hausdorff topological group since AK;S is such a group and OK;S is a closed subgroup. Using the continuous projection to the S-factor, we get a continuous surjection AK;S =OK;S ( Kv)=OK;S Y v S 2 to a compact group, and the kernel is surjected onto by the compact group v S Ov, so the kernel is itself Q 62 compact. Hence, to infer compactness of AK;S =OK;S we use the rather general: Lemma 1.1. Let G be a locally compact Hausdorff topological group, and let H be a closed subgroup.