Invitation to Higher Local Fields
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Local Class Field Theory Via Lubin-Tate Theory and Applications
Local Class Field Theory via Lubin-Tate Theory and Applications Misja F.A. Steinmetz Supervisor: Prof. Fred Diamond June 2016 Abstract In this project we will introduce the study of local class field theory via Lubin-Tate theory following Yoshida [Yos08]. We explain how to construct the Artin map for Lubin-Tate extensions and we will show this map gives an isomorphism onto the Weil group of the maximal Lubin-Tate extension of our local field K: We will, furthermore, state (without proof) the other results needed to complete a proof of local class field theory in the classical sense. At the end of the project, we will look at a result from a recent preprint of Demb´el´e,Diamond and Roberts which gives an 1 explicit description of a filtration on H (GK ; Fp(χ)) for K a finite unramified extension of Qp and × χ : GK ! Fp a character. Using local class field theory, we will prove an analogue of this result for K a totally tamely ramified extension of Qp: 1 Contents 1 Introduction 3 2 Local Class Field Theory5 2.1 Formal Groups..........................................5 2.2 Lubin-Tate series.........................................6 2.3 Lubin-Tate Modules.......................................7 2.4 Lubin-Tate Extensions for OK ..................................8 2.5 Lubin-Tate Groups........................................9 2.6 Generalised Lubin-Tate Extensions............................... 10 2.7 The Artin Map.......................................... 11 2.8 Local Class Field Theory.................................... 13 1 3 Applications: Filtration on H (GK ; Fp(χ)) 14 3.1 Definition of the filtration.................................... 14 3.2 Computation of the jumps in the filtration.......................... -
Arithmetic Duality Theorems
Arithmetic Duality Theorems Second Edition J.S. Milne Copyright c 2004, 2006 J.S. Milne. The electronic version of this work is licensed under a Creative Commons Li- cense: http://creativecommons.org/licenses/by-nc-nd/2.5/ Briefly, you are free to copy the electronic version of the work for noncommercial purposes under certain conditions (see the link for a precise statement). Single paper copies for noncommercial personal use may be made without ex- plicit permission from the copyright holder. All other rights reserved. First edition published by Academic Press 1986. A paperback version of this work is available from booksellers worldwide and from the publisher: BookSurge, LLC, www.booksurge.com, 1-866-308-6235, [email protected] BibTeX information @book{milne2006, author={J.S. Milne}, title={Arithmetic Duality Theorems}, year={2006}, publisher={BookSurge, LLC}, edition={Second}, pages={viii+339}, isbn={1-4196-4274-X} } QA247 .M554 Contents Contents iii I Galois Cohomology 1 0 Preliminaries............................ 2 1 Duality relative to a class formation . ............. 17 2 Localfields............................. 26 3 Abelianvarietiesoverlocalfields.................. 40 4 Globalfields............................. 48 5 Global Euler-Poincar´echaracteristics................ 66 6 Abelianvarietiesoverglobalfields................. 72 7 An application to the conjecture of Birch and Swinnerton-Dyer . 93 8 Abelianclassfieldtheory......................101 9 Otherapplications..........................116 AppendixA:Classfieldtheoryforfunctionfields............126 -
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. -
Algebraic Number Theory II
Algebraic number theory II Uwe Jannsen Contents 1 Infinite Galois theory2 2 Projective and inductive limits9 3 Cohomology of groups and pro-finite groups 15 4 Basics about modules and homological Algebra 21 5 Applications to group cohomology 31 6 Hilbert 90 and Kummer theory 41 7 Properties of group cohomology 48 8 Tate cohomology for finite groups 53 9 Cohomology of cyclic groups 56 10 The cup product 63 11 The corestriction 70 12 Local class field theory I 75 13 Three Theorems of Tate 80 14 Abstract class field theory 83 15 Local class field theory II 91 16 Local class field theory III 94 17 Global class field theory I 97 0 18 Global class field theory II 101 19 Global class field theory III 107 20 Global class field theory IV 112 1 Infinite Galois theory An algebraic field extension L/K is called Galois, if it is normal and separable. For this, L/K does not need to have finite degree. For example, for a finite field Fp with p elements (p a prime number), the algebraic closure Fp is Galois over Fp, and has infinite degree. We define in this general situation Definition 1.1 Let L/K be a Galois extension. Then the Galois group of L over K is defined as Gal(L/K) := AutK (L) = {σ : L → L | σ field automorphisms, σ(x) = x for all x ∈ K}. But the main theorem of Galois theory (correspondence between all subgroups of Gal(L/K) and all intermediate fields of L/K) only holds for finite extensions! To obtain the correct answer, one needs a topology on Gal(L/K): Definition 1.2 Let L/K be a Galois extension. -
Questions and Remarks to the Langlands Program
Questions and remarks to the Langlands program1 A. N. Parshin (Uspekhi Matem. Nauk, 67(2012), n 3, 115-146; Russian Mathematical Surveys, 67(2012), n 3, 509-539) Introduction ....................................... ......................1 Basic fields from the viewpoint of the scheme theory. .............7 Two-dimensional generalization of the Langlands correspondence . 10 Functorial properties of the Langlands correspondence. ................12 Relation with the geometric Drinfeld-Langlands correspondence . 16 Direct image conjecture . ...................20 A link with the Hasse-Weil conjecture . ................27 Appendix: zero-dimensional generalization of the Langlands correspondence . .................30 References......................................... .....................33 Introduction The goal of the Langlands program is a correspondence between representations of the Galois groups (and their generalizations or versions) and representations of reductive algebraic groups. The starting point for the construction is a field. Six types of the fields are considered: three types of local fields and three types of global fields [L3, F2]. The former ones are the following: 1) finite extensions of the field Qp of p-adic numbers, the field R of real numbers and the field C of complex numbers, arXiv:1307.1878v1 [math.NT] 7 Jul 2013 2) the fields Fq((t)) of Laurent power series, where Fq is the finite field of q elements, 3) the field of Laurent power series C((t)). The global fields are: 4) fields of algebraic numbers (= finite extensions of the field Q of rational numbers), 1I am grateful to R. P. Langlands for very useful conversations during his visit to the Steklov Mathematical institute of the Russian Academy of Sciences (Moscow, October 2011), to Michael Harris and Ulrich Stuhler, who answered my sometimes too naive questions, and to Ilhan˙ Ikeda˙ who has read a first version of the text and has made several remarks. -
Abstracts Julio Cesar Bueno De Andrade (University of Exeter, UK): Universality of L-Functions in Function Fields
NEW DEVELOPMENTS IN THE THEORY OF MODULAR FORMS OVER FUNCTION FIELDS CRM PISA, NOVEMBER 5{9, 2018 Titles and Abstracts Julio Cesar Bueno De Andrade (University of Exeter, UK): Universality of L-functions in Function Fields. In this talk, I will discuss Dirichlet L-functions associated with Dirich- let characters in Fq[x] and will prove that they are universal. That is, given a modulus of high enough degree, L-functions with characters to this modulus can be found that approximate any given nonvanishing analytic function arbitrarily closely. This is the function field analogue of Voronin's universality theorem. Dirk Basson (University of Stellenbosch, South Africa): Hecke operators in higher rank. Recently, F. Breuer, R. Pink and I have made available preprints on a foundational theory for Drinfeld modular form of rank greater than 2. In this talk I wish to give an overview of this theory with a focus on the Hecke operators that act on them. Gunther Cornelissen (Utrecht University, Netherlands): Count- ing in algebraic groups in positive characteristic. Our starting point are the well-known formulae for the number of places of a given degree in a global function field and for the order of groups of Lie type over finite fields. We consider these as results about the action of Frobenius on jacobians and on the additive group, and in- vestigate what happens for general endomorphisms of general algebraic groups. New p-adic phenomena appear, zeta functions are no longer rational in general, and the analogue of the Prime Number Theorem can fail. (Joint work with Jakub Byszewski and Marc Houben.) Madeline Locus Dawsey (Emory University, USA): Higher Width Moonshine. -
Drinfeld Modules
Drinfeld modules Jared Weinstein November 8, 2017 1 Motivation Drinfeld introduced his so-called elliptic modules in an important 1973 paper1 of the same title. He introduces this paper by observing the unity of three phenomena that appear in number theory: 1. The theory of cyclotomic extensions of Q, and class field theory over Q. 2. The theory of elliptic curves with complex multiplication, relative to an imaginary quadratic field. 3. The theory of elliptic curves in the large, over Q. To these, Drinfeld added a fourth: 4. The theory of Drinfeld modules over a function field. Thus, Drinfeld modules are some kind of simultaneous generalization of groups of roots of unity (which are rank 1), but also of elliptic curves (which are rank 2, in the appropriate sense). Furthermore, Drinfeld modules can be any rank whatsoever; there is no structure that we know of which is an analogue of a rank 3 Drinfeld module over Q. In later work, Drinfeld extended his notion to a rather more general gad- get called a shtuka2. Later, Laurent Lafforgue used the cohomology of moduli spaces of shtukas to prove the Langlands conjectures for GL(n), generalizing 1Try not to think too hard about the fact that Drinfeld was 20 years old that year. 2Russian slang for \thingy", from German St¨uck, \piece". 1 what Drinfeld had done for n = 2, and receiving a Fields medal in 2002 for those efforts. Whereas the Langlands conjectures are still wide open for number fields, even for GL(2) and even for Q! Before stating the definition of a Drinfeld module, it will be helpful to review items (1)-(3) above and highlight the common thread. -
T-Motives: Hodge Structures, Transcendence and Other Motivic Aspects
t-motives: Hodge structures, transcendence and other motivic aspects G. B¨ockle, D. Goss, U. Hartl, M. Papanikolas December 6, 2009 1 Overview of the field Drinfeld in 1974, in his seminal paper [10], revolutionized the contribution to arithmetic of the area of global function fields. He introduced a function field analog of elliptic curves over number fields. These analogs are now called Drinfeld modules. For him and for many subsequent developments in the theory of automorphic forms over function fields, their main use was in the exploration of the global Langlands conjecture over function fields. One of its predictions is a correspondence between automorphic forms and Galois representations. The deep insight of Drinfeld was that the moduli spaces of Drinfeld modules can be assembled in a certain tower such that the corresponding direct limit of the associated `-adic cohomologies would be an automorphic representation which at the same time carries a Galois action. This would allow him to realize the correspondence conjectured by Langlands in geometry. Building on this, in [11] Drinfeld himself proved the global Langlands' correspondence for function fields for GL2 and later in [18] Lafforgue obtained the result for all GLn. In a second direction, the analogy of Drinfeld modules with elliptic curves over number fields made them interesting objects to be studied on their own right. One could study torsion points and Galois representations, one could define cohomology theories such as de Rham or Betti cohomology and thus investigate their periods as well as transcendence questions. A main advance in this direction is the introduction of t-motives by Anderson in [1]. -
Special L-Values of Drinfeld Modules
Annals of Mathematics 175 (2012), 369{391 http://dx.doi.org/10.4007/annals.2012.175.1.10 Special L-values of Drinfeld modules By Lenny Taelman Abstract We state and prove a formula for a certain value of the Goss L-function of a Drinfeld module. This gives characteristic-p-valued function field ana- logues of the class number formula and of the Birch and Swinnerton-Dyer conjecture. The formula and its proof are presented in an entirely self- contained fashion. Contents 1. Introduction and statement of the theorem 369 2. Nuclear operators and determinants 376 3. A trace formula 381 4. Ratio of co-volumes 384 5. Proof of the main result 388 References 389 1. Introduction and statement of the theorem Let k be a finite field of q elements. For a finite k[t]-module M we define jMj 2 k[T ] to be the characteristic polynomial of the endomorphism t of the k-vector space M, so jMj = detk[T ] T − t M : ∼ Q In other words, if M = ⊕ik[t]=fi(t) with the fi monic then jMj = i fi(T ). We will treat the invariant jMj of the finite k[t]-module M as an analogue of the cardinality #A of a finite abelian group A. Let R be the integral closure of k[t] in a finite extension K of the field k(t) of rational functions. Consider in the Laurent series field k((T −1)) the infinite sum 1 X ; jR=Ij I where I ranges over all the nonzero ideals of R. -
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. -
Valuation Rings Definition 1 ([AK2017, 23.1]). a Discrete Valuation on a Field K Is a Surjective Function V : K × → Z Such Th
Valuation rings Definition 1 ([AK2017, 23.1]). A discrete valuation on a field K is a surjective function × v : K ! Z such that for every x; y 2 K×, (1) v(xy) = v(x) + v(y), (2) v(x + y) ≥ minfv(x); v(y)g if x 6= −y. A = fx 2 K j v(x) ≥ 0g is called the discrete valuation ring of v.A discrete valuation ring, or DVR, is a ring which is a valuation ring of a discrete valuation. Example 2. P1 n (1) The field C((t)) = f n=N ant j N 2 N; an 2 Cg of Laurent series without an essential singularity at t = 0 is the protypical example of a discrete va- P1 n lution ring. The valuation v : C((t)) ! Z sends a series f(t) = n=N ant with aN 6= 0 to N. That is, v(f) is the order of the zero of f at 0 2 C (or minus the order of the pole) when f is considered as a meromorphic function on some neighbourhood of 0 2 C. This can be generalised to any field k by defining 1 X n k((t)) = f ant j N 2 N; an 2 kg: n=N (2) Moreover, if A contains a field k in such a way that k ! A=hπi is an isomorphism for π 2 A with v(π) = 1, then one can show that there is an induced inclusion of fields K ⊂ k((t)), and the valuation of K is induced by that of k((t)). -
Computing the Characteristic Polynomial of a Finite Rank Two Drinfeld Module
Computing the Characteristic Polynomial of a Finite Rank Two Drinfeld Module Yossef Musleh Éric Schost Cheriton School of Computer Science Cheriton School of Computer Science University of Waterloo University of Waterloo Waterloo, Ontario, Canada Waterloo, Ontario, Canada [email protected] [email protected] Abstract Rank two Drinfeld modules enjoy remarkable similarities with Motivated by finding analogues of elliptic curve point counting elliptic curves: analogues exist of good reduction, complex multi- techniques, we introduce one deterministic and two new Monte plication, etc. Based in part on these similarities, Drinfeld modules Carlo randomized algorithms to compute the characteristic poly- have recently started being considered under the algorithmic view- nomial of a finite rank-two Drinfeld module. We compare their point. For instance, they have been proved to be unsuitable for usual asymptotic complexity to that of previous algorithms given by forms of public key cryptography [34]; they have also been used to Gekeler, Narayanan and Garai-Papikian and discuss their practical design several polynomial factorization algorithms [7, 29, 30, 38]; behavior. In particular, we find that all three approaches represent recent work by Garai and Papikian discusses the computation of either an improvement in complexity or an expansion of the pa- their endomorphism rings [9]. Our goal is to study in detail the rameter space over which the algorithm may be applied. Some complexity of computing the characteristic polynomial of a rank experimental results are also presented. two Drinfeld module over a finite field. A fundamental object attached to an elliptic curve E defined CCS Concepts over a finite field Fq is its Frobenius endomorphism π : ¹x;yº 7! ¹xq;yq º; it is known to satisfy a degree-two relation with integer • Computing methodologies → Symbolic and algebraic algo- coefficients called its characteristic polynomial.