On the History of Artin's L-Functions and Conductors Seven Letters from Artin to Hasse in the Year 1930
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Artin L-Functions
Artin L-functions Charlotte Euvrard Laboratoire de Mathématiques de Besançon January 10, 2014 Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 1 / 12 Definition L=K Galois extension of number fields G = Gal(L=K ) its Galois group Let (r;V ) be a representation of G and c the associated character. Let p be a prime ideal in the ring of integers OK of K , we denote by : Ip the inertia group jp the Frobenius automorphism I V p = fv 2 V : 8i 2 Ip; r(i)(v) = vg Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 2 / 12 1 In the particular case K = Q : L(s;c;L=Q) = ∏ det(Id − p−sj ;V Ip ) p2Z p Definition Definition The Artin L-function is defined by : 1 L(s; ;L=K ) = ; Re(s) > 1; c ∏ −s I det(Id − N(p) jp;V p ) p⊂OK where 8 −s < det(Id − N(p) r(jp)) if p is unramified −s Ip −s I det(Id−N(p) jp;V ) = det(Id − N(p) r~(jp)) if p is ramified and V p 6= f0g : 1 otherwise and 1 r~(s¯) = r(js) jI j ∑ p j2Ip Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 3 / 12 Definition Definition The Artin L-function is defined by : 1 L(s; ;L=K ) = ; Re(s) > 1; c ∏ −s I det(Id − N(p) jp;V p ) p⊂OK where 8 −s < det(Id − N(p) r(jp)) if p is unramified −s Ip −s I det(Id−N(p) jp;V ) = det(Id − N(p) r~(jp)) if p is ramified and V p 6= f0g : 1 otherwise and 1 r~(s¯) = r(js) jI j ∑ p j2Ip 1 In the particular case K = Q : L(s;c;L=Q) = ∏ det(Id − p−sj ;V Ip ) p2Z p Charlotte Euvrard (LMB) Artin L-functions Atelier PARI/GP 3 / 12 Sum and Euler product Proposition Let d and d0 be the degrees of the representations (r;V ) and (r~;V Ip ). -
Bibliography
Bibliography [1] Emil Artin. Galois Theory. Dover, second edition, 1964. [2] Michael Artin. Algebra. Prentice Hall, first edition, 1991. [3] M. F. Atiyah and I. G. Macdonald. Introduction to Commutative Algebra. Addison Wesley, third edition, 1969. [4] Nicolas Bourbaki. Alg`ebre, Chapitres 1-3.El´ements de Math´ematiques. Hermann, 1970. [5] Nicolas Bourbaki. Alg`ebre, Chapitre 10.El´ements de Math´ematiques. Masson, 1980. [6] Nicolas Bourbaki. Alg`ebre, Chapitres 4-7.El´ements de Math´ematiques. Masson, 1981. [7] Nicolas Bourbaki. Alg`ebre Commutative, Chapitres 8-9.El´ements de Math´ematiques. Masson, 1983. [8] Nicolas Bourbaki. Elements of Mathematics. Commutative Algebra, Chapters 1-7. Springer–Verlag, 1989. [9] Henri Cartan and Samuel Eilenberg. Homological Algebra. Princeton Math. Series, No. 19. Princeton University Press, 1956. [10] Jean Dieudonn´e. Panorama des mat´ematiques pures. Le choix bourbachique. Gauthiers-Villars, second edition, 1979. [11] David S. Dummit and Richard M. Foote. Abstract Algebra. Wiley, second edition, 1999. [12] Albert Einstein. Zur Elektrodynamik bewegter K¨orper. Annalen der Physik, 17:891–921, 1905. [13] David Eisenbud. Commutative Algebra With A View Toward Algebraic Geometry. GTM No. 150. Springer–Verlag, first edition, 1995. [14] Jean-Pierre Escofier. Galois Theory. GTM No. 204. Springer Verlag, first edition, 2001. [15] Peter Freyd. Abelian Categories. An Introduction to the theory of functors. Harper and Row, first edition, 1964. [16] Sergei I. Gelfand and Yuri I. Manin. Homological Algebra. Springer, first edition, 1999. [17] Sergei I. Gelfand and Yuri I. Manin. Methods of Homological Algebra. Springer, second edition, 2003. [18] Roger Godement. Topologie Alg´ebrique et Th´eorie des Faisceaux. -
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. -
Tōhoku Rick Jardine
INFERENCE / Vol. 1, No. 3 Tōhoku Rick Jardine he publication of Alexander Grothendieck’s learning led to great advances: the axiomatic description paper, “Sur quelques points d’algèbre homo- of homology theory, the theory of adjoint functors, and, of logique” (Some Aspects of Homological Algebra), course, the concepts introduced in Tōhoku.5 Tin the 1957 number of the Tōhoku Mathematical Journal, This great paper has elicited much by way of commen- was a turning point in homological algebra, algebraic tary, but Grothendieck’s motivations in writing it remain topology and algebraic geometry.1 The paper introduced obscure. In a letter to Serre, he wrote that he was making a ideas that are now fundamental; its language has with- systematic review of his thoughts on homological algebra.6 stood the test of time. It is still widely read today for the He did not say why, but the context suggests that he was clarity of its ideas and proofs. Mathematicians refer to it thinking about sheaf cohomology. He may have been think- simply as the Tōhoku paper. ing as he did, because he could. This is how many research One word is almost always enough—Tōhoku. projects in mathematics begin. The radical change in Gro- Grothendieck’s doctoral thesis was, by way of contrast, thendieck’s interests was best explained by Colin McLarty, on functional analysis.2 The thesis contained important who suggested that in 1953 or so, Serre inveigled Gro- results on the tensor products of topological vector spaces, thendieck into working on the Weil conjectures.7 The Weil and introduced mathematicians to the theory of nuclear conjectures were certainly well known within the Paris spaces. -
Part III Essay on Serre's Conjecture
Serre’s conjecture Alex J. Best June 2015 Contents 1 Introduction 2 2 Background 2 2.1 Modular forms . 2 2.2 Galois representations . 6 3 Obtaining Galois representations from modular forms 13 3.1 Congruences for Ramanujan’s t function . 13 3.2 Attaching Galois representations to general eigenforms . 15 4 Serre’s conjecture 17 4.1 The qualitative form . 17 4.2 The refined form . 18 4.3 Results on Galois representations associated to modular forms 19 4.4 The level . 21 4.5 The character and the weight mod p − 1 . 22 4.6 The weight . 24 4.6.1 The level 2 case . 25 4.6.2 The level 1 tame case . 27 4.6.3 The level 1 non-tame case . 28 4.7 A counterexample . 30 4.8 The proof . 31 5 Examples 32 5.1 A Galois representation arising from D . 32 5.2 A Galois representation arising from a D4 extension . 33 6 Consequences 35 6.1 Finiteness of classes of Galois representations . 35 6.2 Unramified mod p Galois representations for small p . 35 6.3 Modularity of abelian varieties . 36 7 References 37 1 1 Introduction In 1987 Jean-Pierre Serre published a paper [Ser87], “Sur les representations´ modulaires de degre´ 2 de Gal(Q/Q)”, in the Duke Mathematical Journal. In this paper Serre outlined a conjecture detailing a precise relationship between certain mod p Galois representations and specific mod p modular forms. This conjecture and its variants have become known as Serre’s conjecture, or sometimes Serre’s modularity conjecture in order to distinguish it from the many other conjectures Serre has made. -
License Or Copyright Restrictions May Apply to Redistribution; See Https
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EMIL ARTIN BY RICHARD BRAUER Emil Artin died of a heart attack on December 20, 1962 at the age of 64. His unexpected death came as a tremendous shock to all who knew him. There had not been any danger signals. It was hard to realize that a person of such strong vitality was gone, that such a great mind had been extinguished by a physical failure of the body. Artin was born in Vienna on March 3,1898. He grew up in Reichen- berg, now Tschechoslovakia, then still part of the Austrian empire. His childhood seems to have been lonely. Among the happiest periods was a school year which he spent in France. What he liked best to remember was his enveloping interest in chemistry during his high school days. In his own view, his inclination towards mathematics did not show before his sixteenth year, while earlier no trace of mathe matical aptitude had been apparent.1 I have often wondered what kind of experience it must have been for a high school teacher to have a student such as Artin in his class. During the first world war, he was drafted into the Austrian Army. After the war, he studied at the University of Leipzig from which he received his Ph.D. in 1921. He became "Privatdozent" at the Univer sity of Hamburg in 1923. -
Right Ideals of a Ring and Sublanguages of Science
RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. -
A Refinement of the Artin Conductor and the Base Change Conductor
A refinement of the Artin conductor and the base change conductor C.-L. Chai and C. Kappen Abstract For a local field K with positive residue characteristic p, we introduce, in the first part of this paper, a refinement bArK of the classical Artin distribution ArK . It takes values in cyclotomic extensions of Q which are unramified at p, and it bisects ArK in the sense that ArK is equal to the sum of bArK and its conjugate distribution. 1 Compared with 2 ArK , the bisection bArK provides a higher resolution on the level of tame ramification. In the second part of this article, we prove that the base change conductor c(T ) of an analytic K-torus T is equal to the value of bArK on the Qp-rational ∗ ∗ Galois representation X (T ) ⊗Zp Qp that is given by the character module X (T ) of T . We hereby generalize a formula for the base change conductor of an algebraic K-torus, and we obtain a formula for the base change conductor of a semiabelian K-variety with potentially ordinary reduction. Contents 1 Introduction 1 2 Notation 4 3 A bisection of the Artin character 5 4 Analytic tori 15 5 The base change conductor 17 6 A formula for the base change conductor 24 7 Application to semiabelian varieties with potentially ordinary reduc- tion 27 References 30 1. Introduction The base change conductor c(A) of a semiabelian variety A over a local field K is an interesting arithmetic invariant; it measures the growth of the volume of the N´eron model of A under base change, cf. -
L-FUNCTIONS Contents Introduction 1 1. Algebraic Number Theory 3 2
L-FUNCTIONS ALEKSANDER HORAWA Contents Introduction1 1. Algebraic Number Theory3 2. Hecke L-functions 10 3. Tate's Thesis: Fourier Analysis in Number Fields and Hecke's Zeta Functions 13 4. Class Field Theory 52 5. Artin L-functions 55 References 79 Introduction These notes, meant as an introduction to the theory of L-functions, form a report from the author's Undergraduate Research Opportunities project at Imperial College London under the supervision of Professor David Helm. The theory of L-functions provides a way to study arithmetic properties of integers (and, more generally, integers of number fields) by first, translating them to analytic properties of certain functions, and then, using the tools and methods of analysis, to study them in this setting. The first indication that the properties of primes could be studied analytically came from Euler, who noticed the factorization (for a real number s > 1): 1 X 1 Y 1 = : ns 1 − p−s n=1 p prime This was later formalized by Riemann, who defined the ζ-function by 1 X 1 ζ(s) = ns n=1 for a complex number s with Re(s) > 1 and proved that it admits an analytic continuation by means of a functional equation. Riemann established a connection between its zeroes and the distribution of prime numbers, and it was also proven that the Prime Number Theorem 1 2 ALEKSANDER HORAWA is equivalent to the fact that no zeroes of ζ lie on the line Re(s) = 1. Innocuous as it may seem, this function is very difficult to study (the Riemann Hypothesis, asserting that the 1 non-trivial zeroes of ζ lie on the line Re(s) = 2 , still remains one of the biggest open problems in mathematics). -
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. -
Introduction. There Are at Least Three Different Problems with Which One Is Confronted in the Study of L-Functions: the Analytic
L-Functions and Automorphic Representations∗ R. P. Langlands Introduction. There are at least three different problems with which one is confronted in the study of L•functions: the analytic continuation and functional equation; the location of the zeroes; and in some cases, the determination of the values at special points. The first may be the easiest. It is certainly the only one with which I have been closely involved. There are two kinds of L•functions, and they will be described below: motivic L•functions which generalize the Artin L•functions and are defined purely arithmetically, and automorphic L•functions, defined by data which are largely transcendental. Within the automorphic L• functions a special class can be singled out, the class of standard L•functions, which generalize the Hecke L•functions and for which the analytic continuation and functional equation can be proved directly. For the other L•functions the analytic continuation is not so easily effected. However all evidence indicates that there are fewer L•functions than the definitions suggest, and that every L•function, motivic or automorphic, is equal to a standard L•function. Such equalities are often deep, and are called reciprocity laws, for historical reasons. Once a reciprocity law can be proved for an L•function, analytic continuation follows, and so, for those who believe in the validity of the reciprocity laws, they and not analytic continuation are the focus of attention, but very few such laws have been established. The automorphic L•functions are defined representation•theoretically, and it should be no surprise that harmonic analysis can be applied to some effect in the study of reciprocity laws. -
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 .