Algebraic Number Theory II
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On the Galois Module Structure of the Square Root of the Inverse Different
University of California Santa Barbara On the Galois Module Structure of the Square Root of the Inverse Different in Abelian Extensions A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Cindy (Sin Yi) Tsang Committee in charge: Professor Adebisi Agboola, Chair Professor Jon McCammond Professor Mihai Putinar June 2016 The Dissertation of Cindy (Sin Yi) Tsang is approved. Professor Jon McCammond Professor Mihai Putinar Professor Adebisi Agboola, Committee Chair March 2016 On the Galois Module Structure of the Square Root of the Inverse Different in Abelian Extensions Copyright c 2016 by Cindy (Sin Yi) Tsang iii Dedicated to my beloved Grandmother. iv Acknowledgements I would like to thank my advisor Prof. Adebisi Agboola for his guidance and support. He has helped me become a more independent researcher in mathematics. I am indebted to him for his advice and help in my research and many other things during my life as a graduate student. I would also like to thank my best friend Tim Cooper for always being there for me. v Curriculum Vitæ Cindy (Sin Yi) Tsang Email: [email protected] Website: https://sites.google.com/site/cindysinyitsang Education 2016 PhD Mathematics (Advisor: Adebisi Agboola) University of California, Santa Barbara 2013 MA Mathematics University of California, Santa Barbara 2011 BS Mathematics (comprehensive option) BA Japanese (with departmental honors) University of Washington, Seattle 2010 Summer program in Japanese language and culture Kobe University, Japan Research Interests Algebraic number theory, Galois module structure in number fields Publications and Preprints (4) Galois module structure of the square root of the inverse different over maximal orders, in preparation. -
Homological Algebra
Homological Algebra Donu Arapura April 1, 2020 Contents 1 Some module theory3 1.1 Modules................................3 1.6 Projective modules..........................5 1.12 Projective modules versus free modules..............7 1.15 Injective modules...........................8 1.21 Tensor products............................9 2 Homology 13 2.1 Simplicial complexes......................... 13 2.8 Complexes............................... 15 2.15 Homotopy............................... 18 2.23 Mapping cones............................ 19 3 Ext groups 21 3.1 Extensions............................... 21 3.11 Projective resolutions........................ 24 3.16 Higher Ext groups.......................... 26 3.22 Characterization of projectives and injectives........... 28 4 Cohomology of groups 32 4.1 Group cohomology.......................... 32 4.6 Bar resolution............................. 33 4.11 Low degree cohomology....................... 34 4.16 Applications to finite groups..................... 36 4.20 Topological interpretation...................... 38 5 Derived Functors and Tor 39 5.1 Abelian categories.......................... 39 5.13 Derived functors........................... 41 5.23 Tor functors.............................. 44 5.28 Homology of a group......................... 45 1 6 Further techniques 47 6.1 Double complexes........................... 47 6.7 Koszul complexes........................... 49 7 Applications to commutative algebra 52 7.1 Global dimensions.......................... 52 7.9 Global dimension of -
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 -
Math 210B. Annihilation of Cohomology of Finite Groups 1. Motivation an Important Fact in the Cohomology Theory of Finite Groups
Math 210B. Annihilation of cohomology of finite groups 1. Motivation An important fact in the cohomology theory of finite groups is that if G is a finite group then Hn(G; M) is killed by #G for any n > 0 and any G-module M. This is not at all obvious from the abstract definition (via injective resolutions, say), nor even from the viewpoint of the explicit \bar resolution" method of computing group cohomology. In this handout we present the proof of this fundamental fact as a marvelous concrete application of the abstract fact that erasable δ-functors are universal (Grothendieck's conceptualization of the \degree-shifting" method that had long been used in earlier versions of cohomology theories throughout algebra and topology). Before we turn to the proof, we record a remarkable consequence: Theorem 1.1. If G is finite and M is a G-module that is finitely generated as an abelian group then Hn(G; M) is finite for all n > 0. Proof. From the bar resolution, we see that Hn(G; M) is finitely generated as an abelian group, as the terms of the bar resolution complex are the abelian groups Map(Gj;M) which are visibly finitely generated (as each Gj is a finite set). Hence, since it is also killed by #G, it must be finite! Actually, the finiteness of such higher cohomologies can be seen in an entirely different way, at the cost of losing the precise information of being annihilated by the specific integer #G (which is very useful in some applications). -
Open Problems: CM-Types Acting on Class Groups
Open problems: CM-types acting on class groups Gabor Wiese∗ October 29, 2003 These are (modified) notes and slides of a talk given in the Leiden number theory seminar, whose general topic are Shimura varieties. The aim of the talk is to formulate a purely number theoretic question (of Brauer-Siegel type), whose (partial) answer might lead to progress on the so-called André-Oort conjecture on special subvarieties of Shimura varieties (thus the link to the seminar). The question ought to be considered as a special case of problem 14 in [EMO], which was proposed by Bas Edixhoven and which we recall now. Let g be a positive integer and let Ag be the moduli space of principally polarized abelian varieties of dimension g. Do there exist C; δ > 0 such that for every CM-point x of Ag (i.e. the corresponding abelian variety has CM) one has G :x C discr(R ) δ; j Q j ≥ j x j where Rx is the centre of the endomorphism ring of the abelian variety correspond- ing to x??? What we will call the Siegel question in the sequel, is obtained by restricting to simple abelian varieties. In the talk I will prove everything that seems to be known on the question up to this moment, namely the case of dimension 2. This was worked out by Bas Edixhoven in [E]. The arguments used are quite “ad hoc” and I will try to point out, where obstacles to gener- alisations seem to lie. The boxes correspond more or less to the slides I used. -
Background on Local Fields and Kummer Theory
MATH 776 BACKGROUND ON LOCAL FIELDS AND KUMMER THEORY ANDREW SNOWDEN Our goal at the moment is to prove the Kronecker{Weber theorem. Before getting to this, we review some of the basic theory of local fields and Kummer theory, both of which will be used constantly throughout this course. 1. Structure of local fields Let K=Qp be a finite extension. We denote the ring of integers by OK . It is a DVR. There is a unique maximal ideal m, which is principal; any generator is called a uniformizer. We often write π for a uniformizer. The quotient OK =m is a finite field, called the residue field; it is often denoted k and its cardinality is often denoted q. Fix a uniformizer π. Every non-zero element x of K can be written uniquely in the form n uπ where u is a unit of OK and n 2 Z; we call n the valuation of x, and often denote it S −n v(x). We thus have K = n≥0 π OK . This shows that K is a direct union of the fractional −n ideals π OK , each of which is a free OK -module of rank one. The additive group OK is d isomorphic to Zp, where d = [K : Qp]. n × ∼ × The decomposition x = uπ shows that K = Z × U, where U = OK is the unit group. This decomposition is non-canonical, as it depends on the choice of π. The exact sequence 0 ! U ! K× !v Z ! 0 is canonical. Choosing a uniformizer is equivalent to choosing a splitting of this exact sequence. -
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. -
Remarks on the Cohomology of Finite Fundamental Groups of 3–Manifolds
Geometry & Topology Monographs 14 (2008) 519–556 519 arXiv version: fonts, pagination and layout may vary from GTM published version Remarks on the cohomology of finite fundamental groups of 3–manifolds SATOSHI TOMODA PETER ZVENGROWSKI Computations based on explicit 4–periodic resolutions are given for the cohomology of the finite groups G known to act freely on S3 , as well as the cohomology rings of the associated 3–manifolds (spherical space forms) M = S3=G. Chain approximations to the diagonal are constructed, and explicit contracting homotopies also constructed for the cases G is a generalized quaternion group, the binary tetrahedral group, or the binary octahedral group. Some applications are briefly discussed. 57M05, 57M60; 20J06 1 Introduction The structure of the cohomology rings of 3–manifolds is an area to which Heiner Zieschang devoted much work and energy, especially from 1993 onwards. This could be considered as part of a larger area of his interest, the degrees of maps between oriented 3– manifolds, especially the existence of degree one maps, which in turn have applications in unexpected areas such as relativity theory (cf Shastri, Williams and Zvengrowski [41] and Shastri and Zvengrowski [42]). References [1,6,7, 18, 19, 20, 21, 22, 23] in this paper, all involving work of Zieschang, his students Aaslepp, Drawe, Sczesny, and various colleagues, attest to his enthusiasm for these topics and the remarkable energy he expended studying them. Much of this work involved Seifert manifolds, in particular, references [1, 6, 7, 18, 20, 23]. Of these, [6, 7, 23] (together with [8, 9]) successfully completed the programme of computing the ring structure H∗(M) for any orientable Seifert manifold M with 1 2 3 3 G := π1(M) infinite. -
Group Cohomology in Lean
BEng Individual Project Imperial College London Department of Computing Group Cohomology in Lean Supervisor: Prof. Kevin Buzzard Author: Anca Ciobanu Second Marker: Prof. David Evans July 10, 2019 Abstract The Lean project is a new open source launched in 2013 by Leonardo de Moura at Microsoft Research Redmond that can be viewed as a programming language specialised in theorem proving. It is suited for formalising mathematical defini- tions and theorems, which can help bridge the gap between the abstract aspect of mathematics and the practical part of informatics. My project focuses on using Lean to formalise some essential notions of group cohomology, such as the 0th and 1st cohomology groups, as well as the long exact sequence. This can be seen as a performance test for the new theorem prover, but most im- portantly as an addition to the open source, as group cohomology has yet to be formalised in Lean. Contents 1 Introduction3 1.1 About theorem provers.......................3 1.2 Lean as a solution.........................4 1.3 Achievements............................4 2 Mathematical Background6 2.1 Basic group theory.........................6 2.2 Normal subgroups and homomorphisms.............7 2.3 Quotient groups..........................7 3 Tutorial in Lean9 3.1 Tactics for proving theorems....................9 3.2 Examples.............................. 10 4 Definitions and Theorems of Group Cohomology 13 4.1 Motivation............................. 13 4.2 Modules............................... 14 4.3 0th cohomology group....................... 14 4.4 First cohomology group...................... 16 5 Group Cohomology in Lean 19 5.1 Exact sequence with H0(G; M) only............... 19 5.2 Long exact sequence........................ 22 6 Evaluation 24 6.1 First impressions on Lean.................... -
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. -
The Cohomology of Automorphism Groups of Free Groups
The cohomology of automorphism groups of free groups Karen Vogtmann∗ Abstract. There are intriguing analogies between automorphism groups of finitely gen- erated free groups and mapping class groups of surfaces on the one hand, and arithmetic groups such as GL(n, Z) on the other. We explore aspects of these analogies, focusing on cohomological properties. Each cohomological feature is studied with the aid of topolog- ical and geometric constructions closely related to the groups. These constructions often reveal unexpected connections with other areas of mathematics. Mathematics Subject Classification (2000). Primary 20F65; Secondary, 20F28. Keywords. Automorphism groups of free groups, Outer space, group cohomology. 1. Introduction In the 1920s and 30s Jakob Nielsen, J. H. C. Whitehead and Wilhelm Magnus in- vented many beautiful combinatorial and topological techniques in their efforts to understand groups of automorphisms of finitely-generated free groups, a tradition which was supplemented by new ideas of J. Stallings in the 1970s and early 1980s. Over the last 20 years mathematicians have been combining these ideas with others motivated by both the theory of arithmetic groups and that of surface mapping class groups. The result has been a surge of activity which has greatly expanded our understanding of these groups and of their relation to many areas of mathe- matics, from number theory to homotopy theory, Lie algebras to bio-mathematics, mathematical physics to low-dimensional topology and geometric group theory. In this article I will focus on progress which has been made in determining cohomological properties of automorphism groups of free groups, and try to in- dicate how this work is connected to some of the areas mentioned above. -
Galois Modules Torsion in Number Fields
Galois modules torsion in Number fields Eliharintsoa RAJAONARIMIRANA ([email protected]) Université de Bordeaux France Supervised by: Prof Boas Erez Université de Bordeaux, France July 2014 Université de Bordeaux 2 Abstract Let E be a number fields with ring of integers R and N be a tame galois extension of E with group G. The ring of integers S of N is an RG−module, so an ZG−module. In this thesis, we study some other RG−modules which appear in the study of the module structure of S as RG− module. We will compute their Hom-representatives in Frohlich Hom-description using Stickelberger’s factorisation and show their triviliaty in the class group Cl(ZG). 3 4 Acknowledgements My appreciation goes first to my supervisor Prof Boas Erez for his guidance and support through out the thesis. His encouragements and criticisms of my work were of immense help. I will not forget to thank the entire ALGANT staffs in Bordeaux and Stellenbosch and all students for their help. Last but not the least, my profond gratitude goes to my family for their prayers and support through out my stay here. 5 6 Contents Abstract 3 1 Introduction 9 1.1 Statement of the problem . .9 1.2 Strategy of the work . 10 1.3 Commutative Algebras . 12 1.4 Completions, unramified and totally ramified extensions . 21 2 Reduction to inertia subgroup 25 2.1 The torsion module RN=E ...................................... 25 2.2 Torsion modules arising from ideals . 27 2.3 Switch to a global cyclotomic field . 30 2.4 Classes of cohomologically trivial modules .