Iwasawa Theory

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Math 205 – Topics in Number Theory (Winter 2015) Professor: Cristian D. Popescu An Introduction to Iwasawa Theory Iwasawa theory grew out of Kenkichi Iwasawa’s efforts (1950s – 1970s) to transfer to characteristic 0 (number fields) the characteristic p (function field) techniques which led A. Weil (1940s) to his interpretation of the zeta function of a smooth projective curve over a finite field in terms of the characteristic polynomial of the action of the geometric Frobenius morphism on its first l-adic étale cohomology group (the l-adic Tate module of its Jacobian.) This transfer of techniques and results has been successful through the efforts of many mathematicians (e.g. Iwasawa, Coates, Greenberg, Mazur, Wiles, Kolyvagin, Rubin etc.), with some caveats: the zeta function has to be replaced by its l-adic interpolations (the so-called l-adic zeta functions of the number field in question) and the l-adic étale cohomology group has to be replaced by a module over a profinite group ring (the so-called Iwasawa algebra). The link between the l-adic zeta functions and the Iwasawa modules is given by the so-called “main conjecture” in Iwasawa theory, in many cases a theorem with strikingly far reaching arithmetic applications. My goal in this course is to explain the general philosophy of Iwasawa theory, state the “main conjecture” in the particular case of Dirichlet L-functions, give a rough outline of its proof and conclude with some concrete arithmetic applications, open problems and new directions of research in this area. Background requirements: solid knowledge of basic algebraic number theory (the Math 204 series). I will review certain topics in algebraic number theory as needed. Bibliography: I will not follow a textbook. However, L. Washington’s “An introduction to cyclotomic fields” and S. Lang’s “Cyclotomic Fields I and II” are good sources of information. .

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Iwasawa Theory and Motivic L-Functions
IWASAWA THEORY AND MOTIVIC L-FUNCTIONS MATTHIAS FLACH to Jean-Pierre Serre Abstract. We illustrate the use of Iwasawa theory in proving cases of the (equivariant) Tamagawa number conjecture. Contents Part 1. Motivic L-functions 2 1. Review of the Tamagawa Number Conjecture 2 2. Limit formulas 5 2.1. Dirichlet L-functions 6 2.2. The Kronecker limit formula 7 2.3. Other Limit Formulas 15 Part 2. Iwasawa Theory 16 3. The Zeta isomorphism of Kato and Fukaya 18 4. Iwasawa theory of imaginary quadratic ¯eld 19 4.1. The main conjecture for imaginary quadratic ¯elds 19 4.2. The explicit reciprocity law 22 5. The cyclotomic deformation and the exponential of Perrin-Riou 31 6. Noncommutative Iwasawa theory 33 References 34 The present paper is a continuation of the survey article [19] on the equivariant Tamagawa number conjecture. We give a presentation of the Iwasawa theory of imaginary quadratic ¯elds from the modern point of Kato and Fukaya [21] and shall take the opportunity to brieﬂy mention developments which have occurred after the publication of [19], most notably the limit formula proved by Gealy [23], and work of Bley 2000 Mathematics Subject Classi¯cation. Primary: 11G40, Secondary: 11R23, 11R33, 11G18. The author was supported by grant DMS-0401403 from the National Science Foundation. 1 2 MATTHIAS FLACH [4], Johnson [27] and Navilarekallu [32]. Unfortunately, we do not say as much about the history of the subject as we had wished. Although originally planned we also did not include a presentation of the full GL2- Iwasawa theory of elliptic modular forms following Kato and Fukaya (see Colmez' paper [14] for a thoroughly p-adic presentation).
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$ P $-Adic Wan-Riemann Hypothesis for $\Mathbb {Z} P $-Towers of Curves
p-adic Wan-Riemann Hypothesis for Zp-towers of curves Roberto Alvarenga Abstract. Our goal in this paper is to investigatefour conjectures, proposed by Daqing Wan, about the stable behavior of a geometric Zp-tower of curves X∞/X. Let hn be the class number of the n-th layer in X∞/X. It is known from Iwasawa theory that there are integers µ(X∞/X),λ(X∞/X) and ν(X∞/X) such that the p-adic valuation vp(hn) n equals to µ(X∞/X)p + λ(X∞/X)n + ν(X∞/X) for n sufficiently large. Let Qp,n be the splitting field (over Qp) of the zeta-function of n-th layer in X∞/X. The p-adic Wan- Riemann Hypothesis conjectures that the extension degree [Qp,n : Qp] goes to infinity as n goes to infinity. After motivating and introducing the conjectures, we prove the p-adic Wan-Riemann Hypothesis when λ(X∞/X) is nonzero. 1. Introduction Intention and scope of this text. Let p be a prime number and Zp be the p-adic integers. Our exposition is meant as a reader friendly introduction to the p-adic Wan- Riemann Hypothesis for geometric Zp-towers of curves. Actually, we shall investigate four conjectures stated by Daqing Wan concerning the stable behavior of zeta-functions attached to a geometric Zp-tower of curves, where these curves are defined over a finite field Fq and q is a power of the prime p. The paper is organized as follows: in this first section we introduce the basic background of this subject and motivate the study of Zp-towers; in the Section 2, we state the four conjectures proposed by Daqing Wan (in particular the p-adic Wan-Riemann Hypothesis for geometric Zp-towers of curves) and re-write those conjectures in terms of some L-functions; in the last section, we prove the main theorem of this article using the T -adic L-function.
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An Introduction to Iwasawa Theory
An Introduction to Iwasawa Theory Notes by Jim L. Brown 2 Contents 1 Introduction 5 2 Background Material 9 2.1 AlgebraicNumberTheory . 9 2.2 CyclotomicFields........................... 11 2.3 InﬁniteGaloisTheory . 16 2.4 ClassFieldTheory .......................... 18 2.4.1 GlobalClassFieldTheory(ideals) . 18 2.4.2 LocalClassFieldTheory . 21 2.4.3 GlobalClassFieldTheory(ideles) . 22 3 SomeResultsontheSizesofClassGroups 25 3.1 Characters............................... 25 3.2 L-functionsandClassNumbers . 29 3.3 p-adicL-functions .......................... 31 3.4 p-adic L-functionsandClassNumbers . 34 3.5 Herbrand’sTheorem . .. .. .. .. .. .. .. .. .. .. 36 4 Zp-extensions 41 4.1 Introduction.............................. 41 4.2 PowerSeriesRings .......................... 42 4.3 A Structure Theorem on ΛO-modules ............... 45 4.4 ProofofIwasawa’stheorem . 48 5 The Iwasawa Main Conjecture 61 5.1 Introduction.............................. 61 5.2 TheMainConjectureandClassGroups . 65 5.3 ClassicalModularForms. 68 5.4 ConversetoHerbrand’sTheorem . 76 5.5 Λ-adicModularForms . 81 5.6 ProofoftheMainConjecture(outline) . 85 3 4 CONTENTS Chapter 1 Introduction These notes are the course notes from a topics course in Iwasawa theory taught at the Ohio State University autumn term of 2006. They are an amalgamation of results found elsewhere with the main two sources being [Wash] and [Skinner]. The early chapters are taken virtually directly from [Wash] with my contribution being the choice of ordering as well as adding details to some arguments. Any mistakes in the notes are mine. There are undoubtably type-o’s (and possibly mathematical errors), please send any corrections to [email protected]. As these are course notes, several proofs are omitted and left for the reader to read on his/her own time.
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Iwasawa Theory: a Climb up the Tower
Iwasawa Theory: A Climb up the Tower Romyar Sharifi Iwasawa theory is an area of number theory that emerged Through such descriptions, Iwasawa theory unveils in- from the foundational work of Kenkichi Iwasawa in the tricate links between algebraic, geometric, and analytic ob- late 1950s and onward. It studies the growth of arithmetic jects of an arithmetic nature. The existence of such links objects, such as class groups, in towers of number fields. is a common theme in many central areas within arith- Its key observation is that a part of this growth exhibits metic geometry. So it is that Iwasawa theory has found a remarkable regularity, which it aims to describe in terms itself a subject of continued great interest. This year’s Ari- of values of meromorphic functions known as 퐿-functions, zona Winter School attracted nearly 300 students hoping such as the Riemann zeta function. to learn about it! The literature on Iwasawa theory is vast and often tech- Romyar Sharifi is a professor of mathematics at UCLA. His email addressis nical, but the underlying ideas possess undeniable beauty. [email protected]. I hope to convey some of this while explaining the origi- Communicated by Notices Associate Editor Daniel Krashen. nal questions of Iwasawa theory and giving a sense of the This material is based upon work supported by the National Science Foundation directions in which the area is heading. under Grant Nos. 1661658 and 1801963. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti/1759 16 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 1 Algebraic Number Theory in which case it has dense image in ℂ.
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Local Homotopy Theory, Iwasawa Theory and Algebraic K–Theory
K(1){local homotopy theory, Iwasawa theory and algebraic K{theory Stephen A. Mitchell? University of Washington, Seattle, WA 98195, USA 1 Introduction The Iwasawa algebra Λ is a power series ring Z`[[T ]], ` a fixed prime. It arises in number theory as the pro-group ring of a certain Galois group, and in homotopy theory as a ring of operations in `-adic complex K{theory. Fur- thermore, these two incarnations of Λ are connected in an interesting way by algebraic K{theory. The main goal of this paper is to explore this connection, concentrating on the ideas and omitting most proofs. Let F be a number field. Fix a prime ` and let F denote the `-adic cyclotomic tower { that is, the extension field formed by1 adjoining the `n-th roots of unity for all n 1. The central strategy of Iwasawa theory is to study number-theoretic invariants≥ associated to F by analyzing how these invariants change as one moves up the cyclotomic tower. Number theorists would in fact consider more general towers, but we will be concerned exclusively with the cyclotomic case. This case can be viewed as analogous to the following geometric picture: Let X be a curve over a finite field F, and form a tower of curves over X by extending scalars to the algebraic closure of F, or perhaps just to the `-adic cyclotomic tower of F. This analogy was first considered by Iwasawa, and has been the source of many fruitful conjectures. As an example of a number-theoretic invariant, consider the norm inverse limit A of the `-torsion part of the class groups in the tower.
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Introduction to Iwasawa Theory
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AN INTRODUCTION to P-ADIC L-FUNCTIONS
AN INTRODUCTION TO p-ADIC L-FUNCTIONS by Joaquín Rodrigues Jacinto & Chris Williams Contents General overview......................................................................... 2 1. Introduction.............................................................................. 2 1.1. Motivation. 2 1.2. The Riemann zeta function. 5 1.3. p-adic L-functions. 6 1.4. Structure of the course.............................................................. 11 Part I: The Kubota–Leopoldt p-adic L-function.................................... 12 2. Measures and Iwasawa algebras.......................................................... 12 2.1. The Iwasawa algebra................................................................ 12 2.2. p-adic analysis and Mahler transforms............................................. 14 2.3. An example: Dirac measures....................................................... 15 2.4. A measure-theoretic toolbox........................................................ 15 3. The Kubota-Leopoldt p-adic L-function. 17 3.1. The measure µa ..................................................................... 17 3.2. Restriction to Zp⇥ ................................................................... 18 3.3. Pseudo-measures.................................................................... 19 4. Interpolation at Dirichlet characters..................................................... 21 4.1. Characters of p-power conductor. 21 4.2. Non-trivial tame conductors........................................................ 23 4.3. Analytic
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MAIN CONJECTURE of IWASAWA THEORY 1. Weil Conjectures 1.1
MAIN CONJECTURE OF IWASAWA THEORY C. S. RAJAN Abstract. 1) Weil conjectures 2) Iwasawa’s theorem on growth of class groups 3) Iwasawa’s construction of p-adic L-functions via Stickelberger elements 4) Main conjecture. 1. Weil Conjectures 1.1. Zeta and L-functions. We recall the Riemann zeta function ζ(s)= n−s (Re(s) > 1). n≥1 X The above series converges absolutely for Re(s) > 1 and defines an analytic function there. ζ(s) admits an analytic continuation to the entire complex plane. The zeta function and it’s generalisations enjoy remarkable analytic properties, which have significant consequences to arithmetic. One of the outstanding problems concerned with the zeta function is the Riemann Hypothesis: Riemann Hypothesis(RH): If ρ is any zero of ζ(s) with Re(ρ) 0 (such zeros are called the non-trivial zeros of ζ(s)), then ≥ Re(ρ) = 1/2. i.e., all the zeros to the right of the line Re(s) = 0 lie on the line Re(s) = 1/2. How does one tackle this conjecture? One idea going back to Hilbert stems from the fact that the eigenvalues of a hermitian (skew-hermitian) matrix are real (purely imaginary). In other words the eigenvalues of a hermitian matrix all lie on a straight line. This leads to the following question: Question 1.1.1. Is it possible to find a skew-hermitian operator acting on some space, such that the zeros of ζ(s + 1/2) with non-negative real part, occur as eigenvalues of this operator? 1991 Mathematics Subject Classification.
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P-Adic Automorphic Forms on Shimura Varieties, by Haruzo Hida, Springer-Verlag, New York, 2004, Xii+390 Pp., US$99.00, ISBN 0-387-20711-2
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 2, April 2007, Pages 291–308 S 0273-0979(06)01131-1 Article electronically published on October 18, 2006 p-Adic automorphic forms on Shimura varieties, by Haruzo Hida, Springer-Verlag, New York, 2004, xii+390 pp., US$99.00, ISBN 0-387-20711-2 Three topics figure prominently in the modern higher arithmetic: zeta-functions, Galois representations, and automorphic forms or, equivalently, representations. The zeta-functions are attached to both the Galois representations and the auto- morphic representations and are the link that joins them. Although by and large abstruse and often highly technical, the subject has many claims on the attention of mathematicians as a whole: the spectacular solution of Fermat’s Last Theorem; concrete conjectures that are both difficult and not completely inaccessible, above all that of Birch and Swinnerton-Dyer; roots in an ancient tradition of the study of algebraic irrationalities; a majestic conceptual architecture with implications not confined to number theory; and great current vigor. Nevertheless, in spite of major results modern arithmetic remains inchoate, with far more conjectures than the- orems. There is no schematic introduction to it that reveals the structure of the conjectures whose proofs are its principal goal and of the methods to be employed, and for good reason. There are still too many uncertainties. I nonetheless found while preparing this review that without forming some notion of the outlines of the final theory, I was quite at sea with the subject and with the book. So ill-equipped as I am in many ways – although not in all – my first, indeed my major, task was to take bearings.
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Iwasawa Theory and Generalizations
Iwasawa theory and generalizations Kazuya Kato Abstract. This is an introduction to Iwasawa theory and its generalizations. We discuss some main conjectures and related subjects. Mathematics Subject Classification (2000). Primary 11R23; Secondary 11G40, 14G10. Keywords. Iwasawa theory, zeta function. Zeta values are infinitely attractive objects. They appear in many areas of mathematics, and also in physics. They lead us to the profound mysteries of mathematics. Iwasawa theory is the best theory at present to understand the arithmetic meaning of zeta values. Classical Iwasawa theory describes the relation between zeta values and ideal class groups. It has been generalized to the study of the relation between zeta values and more general arithmetic objects (rational points and Selmer groups of elliptic curves, Galois representations, Galois cohomology groups, …). Diagram- matically, we have zeta values ←→ ideal class groups, Selmer groups,.... (analytic objects) (arithmetic objects) The mysterious point is that analytic objects and arithmetic objects are connected in spite of the vast distance between their innate natures. Via zeta values, arithmetic, algebra, analysis, geometry intersect. Deep and unexpected problems arise there. The aim of this paper is to review some results, methods, and problems in Iwasawa theory and its generalizations. The author regrets that he cannot cover many important studies in this field. The organization of this paper is as follows. In §1, we describe history. In §2, we describe the cyclotomic Iwasawa theory of elliptic curves comparing it with classical Iwasawa theory. In §3, we present non-commutative Iwasawa theory. The author is very grateful to Professor John Coates for advice. 1. Overview 1.1.
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Lectures on the Iwasawa Theory of Elliptic Curves
LECTURES ON THE IWASAWA THEORY OF ELLIPTIC CURVES CHRISTOPHER SKINNER Abstract. These are a preliminary set ot notes for the author's lectures for the 2018 Arizona Winter School on Iwasawa Theory. Contents 1. Introduction 2 2. Selmer groups 3 2.1. Selmer groups of elliptic curves 3 2.2. Bloch{Kato Selmer groups 9 2.3. Selmer structures 12 3. Iwasawa modules for elliptic curves 13 3.1. The extension F1=F 13 3.2. Selmer groups over F1 15 3.3. S?(E=F1) as a Λ-module 17 3.4. Control theorems 19 4. Main Conjectures 22 4.1. p-adic L-functions 23 4.2. The Main Conjectures 27 4.3. Main Conjectures without L-functions 28 5. Theorems and ideas of their proofs 30 5.1. Cyclotomic Main Conjectures: the ordinary case 30 ac 5.2. The Main Conjectures for SGr(E=K1) and SBDP(E=K1) 32 5.3. Cyclotomic Main Conjectures: the supersingular case 34 5.4. Perrin-Riou's Heegner point Main Conjecture 35 6. Arithmetic consequences 36 1 2 CHRISTOPHER SKINNER 6.1. Results when L(E; 1) 6= 0 36 6.2. Results when L(E; 1) = 0 36 6.3. Results when ords=1L(E; s) = 1 37 6.4. Converses to Gross{Zagier/Kolyvagin 39 References 40 1. Introduction Iwasawa theory was introduced around 1960 in the context of class groups of cyclotomic and other Zp-extensions of number ﬁelds. The Main Conjecture of Iwasawa theory proposed a re- markable connection between the p-adic L-functions of Kubota and Leopoldt and these class groups [19, x1], [12, x5], including among its consequences certain reﬁned class number formulas for values of Dirichlet L-functions.
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The Emerging P-Adic Langlands Programme
Proceedings of the International Congress of Mathematicians Hyderabad, India, 2010 The emerging p-adic Langlands programme Christophe Breuil ∗ Abstract. We give a brief overview of some aspects of the p-adic and modulo p Langlands programmes. Mathematics Subject Classification (2000). Primary 11S80; Secondary 22D12. Keywords. p-adic Langlands programme, p-adic Hodge theory, GL2(Qp), ('; Γ)-modules, completed cohomology. 1. Introduction Fix p a prime number and Qp an algebraic closure of the field of p-adic numbers Qp. Let ` 6= p be another prime number and Q` an algebraic closure of Q`. If F is a field which is a finite extension of Qp and n a positive integer, the celebrated local Langlands programme for GLn ([48], [37], [38]) establishes a \natural" 1 − 1 cor- respondence between certain Q`-linear continuous representations ρ of the Galois group Gal(Qp=F ) on n-dimensional Q`-vector spaces and certain Q`-linear locally constant (or smooth) irreducible representations π of GLn(F ) on (usually infinite dimensional) Q`-vector spaces. This local correspondence is moreover compatible with reduction modulo ` ([68]) and with cohomology ([49], [23], [37]). By \compat- ible with cohomology", we mean here that there exist towers of algebraic (Shimura) varieties (S(K))K over F of dimension d indexed by compact open subgroups K of GLn(F ) on which GLn(F ) acts on the right and such that the natural action of GLn(F ) × Gal(Qp=F ) on the inductive limit of `-adic étalecohomology groups: d lim H (S(K) ×F p; `) (1) −! ét Q Q K makes it a direct sum of representations π ⊗ ρ where ρ matches π by the previous local correspondence.
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