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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. -
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. -
Arxiv:0906.3146V1 [Math.NT] 17 Jun 2009
Λ-RINGS AND THE FIELD WITH ONE ELEMENT JAMES BORGER Abstract. The theory of Λ-rings, in the sense of Grothendieck’s Riemann– Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce Λ-algebraic geometry. We show that Λ-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry. Introduction Many writers have mused about algebraic geometry over deeper bases than the ring Z of integers. Although there are several, possibly unrelated reasons for this, here I will mention just two. The first is that the combinatorial nature of enumer- ation formulas in linear algebra over finite fields Fq as q tends to 1 suggests that, just as one can work over all finite fields simultaneously by using algebraic geome- try over Z, perhaps one could bring in the combinatorics of finite sets by working over an even deeper base, one which somehow allows q = 1. It is common, follow- ing Tits [60], to call this mythical base F1, the field with one element. (See also Steinberg [58], p. 279.) The second purpose is to prove the Riemann hypothesis. With the analogy between integers and polynomials in mind, we might hope that Spec Z would be a kind of curve over Spec F1, that Spec Z ⊗F1 Z would not only make sense but be a surface bearing some kind of intersection theory, and that we could then mimic over Z Weil’s proof [64] of the Riemann hypothesis over function fields.1 Of course, since Z is the initial object in the category of rings, any theory of algebraic geometry over a deeper base would have to leave the usual world of rings and schemes. -
The Work of Pierre Deligne
THE WORK OF PIERRE DELIGNE W.T. GOWERS 1. Introduction Pierre Deligne is indisputably one of the world's greatest mathematicians. He has re- ceived many major awards, including the Fields Medal in 1978, the Crafoord Prize in 1988, the Balzan Prize in 2004, and the Wolf Prize in 2008. While one never knows who will win the Abel Prize in any given year, it was virtually inevitable that Deligne would win it in due course, so today's announcement is about as small a surprise as such announcements can be. This is the third time I have been asked to present to a general audience the work of the winner of the Abel Prize, and my assignment this year is by some way the most difficult of the three. Two years ago, I talked about John Milnor, whose work in geometry could be illustrated by several pictures. Last year, the winner was Endre Szemer´edi,who has solved several problems with statements that are relatively simple to explain (even if the proofs were very hard). But Deligne's work, though it certainly has geometrical aspects, is not geometrical in a sense that lends itself to pictorial explanations, and the statements of his results are far from elementary. So I am forced to be somewhat impressionistic in my description of his work and to spend most of my time discussing the background to it rather than the work itself. 2. The Ramanujan conjecture One of the results that Deligne is famous for is solving a conjecture of Ramanujan. This conjecture is not the obvious logical starting point for an account of Deligne's work, but its solution is the most concrete of his major results and therefore the easiest to explain. -
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. -
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. -
Multiplicative Reduction and the Cyclotomic Main Conjecture for Gl2
Pacific Journal of Mathematics MULTIPLICATIVE REDUCTION AND THE CYCLOTOMIC MAIN CONJECTURE FOR GL2 CHRISTOPHER SKINNER Volume 283 No. 1 July 2016 PACIFIC JOURNAL OF MATHEMATICS Vol. 283, No. 1, 2016 dx.doi.org/10.2140/pjm.2016.283.171 MULTIPLICATIVE REDUCTION AND THE CYCLOTOMIC MAIN CONJECTURE FOR GL2 CHRISTOPHER SKINNER We show that the cyclotomic Iwasawa–Greenberg main conjecture holds for a large class of modular forms with multiplicative reduction at p, extending previous results for the good ordinary case. In fact, the multiplicative case is deduced from the good case through the use of Hida families and a simple Fitting ideal argument. 1. Introduction The cyclotomic Iwasawa–Greenberg main conjecture was established in[Skinner and Urban 2014], in combination with work of Kato[2004], for a large class of newforms f 2 Sk.00.N// that are ordinary at an odd prime p - N, subject to k ≡ 2 .mod p − 1/ and certain conditions on the mod p Galois representation associated with f . The purpose of this note is to extend this result to the case where p j N (in which case k is necessarily equal to 2). P1 n Recall that the coefficients an of the q-expansion f D nD1 anq of f at the cusp at infinity (equivalently, the Hecke eigenvalues of f ) are algebraic integers that generate a finite extension Q. f / ⊂ C of Q. Let p be an odd prime and let L be a finite extension of the completion of Q. f / at a chosen prime above p (equivalently, let L be a finite extension of Qp in a fixed algebraic closure Qp of Qp that contains the image of a chosen embedding Q. -
Annual Report 2017-2018
Annual Review 2017 | 2018 ONTENTS C 1 Overview 1 2 Profile 4 3 Research 6 4 Events 9 5 Personnel 13 6 Mentoring 17 7 Structures 18 APPENDICES R1 Highlighted Papers 20 R2 Complete List of Papers 23 E1 HIMR-run Events 29 E2 HIMR-sponsored Events 31 E3 Focused Research Events 39 E4 Future Events 54 P1 Fellows Joining in 2017|2018 59 P2 Fellows Leaving since September 2017 60 P3 Fellows Moving with 3-year Extensions 62 P4 Future Fellows 63 M1 Mentoring Programme 64 1. Overview This has been another excellent year for the Heilbronn Institute, which is now firmly established as a major national mathematical research centre. HIMR has developed a strong brand and is increasingly influential in the UK mathematics community. There is currently an outstanding cohort of Heilbronn Research Fellows doing first-rate research. Recruitment of new Fellows has been most encouraging, as is the fact that many distinguished academic mathematicians continue to work with the Institute. The research culture at HIMR is excellent. Members have expressed a high level of satisfaction. This is especially the case with the Fellows, many of whom have chosen to continue their relationships with the Institute. Our new Fellows come from leading mathematics departments and have excellent academic credentials. Those who left have moved to high-profile groups, including several to permanent academic positions. We currently have 29 Fellows, hosted by 6 universities. We are encouraged by the fact that of the 9 Fellows joining us this year, 5 are women. The achievements of our Fellows this year again range from winning prestigious prizes to publishing in the elite mathematical journals and organising major mathematical meetings. -
Viewed As a Vector Space Over Itself) Equipped with the Gfv -Action Induced by Ψ
New York Journal of Mathematics New York J. Math. 27 (2021) 437{467. On Bloch{Kato Selmer groups and Iwasawa theory of p-adic Galois representations Matteo Longo and Stefano Vigni Abstract. A result due to R. Greenberg gives a relation between the cardinality of Selmer groups of elliptic curves over number fields and the characteristic power series of Pontryagin duals of Selmer groups over cyclotomic Zp-extensions at good ordinary primes p. We extend Green- berg's result to more general p-adic Galois representations, including a large subclass of those attached to p-ordinary modular forms of weight at least 4 and level Γ0(N) with p - N. Contents 1. Introduction 437 2. Galois representations 440 3. Selmer groups 446 4. Characteristic power series 449 Γ 5. Relating SelBK(A=F ) and S 450 6. Main result 464 References 465 1. Introduction A classical result of R. Greenberg ([9, Theorem 4.1]) establishes a relation between the cardinality of Selmer groups of elliptic curves over number fields and the characteristic power series of Pontryagin duals of Selmer groups over cyclotomic Zp-extensions at good ordinary primes p. Our goal in this paper is to extend Greenberg's result to more general p-adic Galois representa- tions, including a large subclass of those coming from p-ordinary modular forms of weight at least 4 and level Γ0(N) with p a prime number such that p - N. This generalization of Greenberg's theorem will play a role in our Received November 11, 2020. 2010 Mathematics Subject Classification. 11R23, 11F80. -
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 . -
[1, 2, 3], Fesenko Defined the Non-Abelian Local Reciprocity
Algebra i analiz St. Petersburg Math. J. Tom 20 (2008), 3 Vol. 20 (2009), No. 3, Pages 407–445 S 1061-0022(09)01054-1 Article electronically published on April 7, 2009 FESENKO RECIPROCITY MAP K. I. IKEDA AND E. SERBEST Dedicated to our teacher Mehpare Bilhan Abstract. In recent papers, Fesenko has defined the non-Abelian local reciprocity map for every totally ramified arithmetically profinite (APF) Galois extension of a given local field K, by extending the work of Hazewinkel and Neukirch–Iwasawa. The theory of Fesenko extends the previous non-Abelian generalizations of local class field theory given by Koch–de Shalit and by A. Gurevich. In this paper, which is research-expository in nature, we give a detailed account of Fesenko’s work, including all the skipped proofs. In a series of very interesting papers [1, 2, 3], Fesenko defined the non-Abelian local reciprocity map for every totally ramified arithmetically profinite (APF) Galois extension of a given local field K by extending the work of Hazewinkel [8] and Neukirch–Iwasawa [15]. “Fesenko theory” extends the previous non-Abelian generalizations of local class field theory given by Koch and de Shalit in [13] and by A. Gurevich in [7]. In this paper, which is research-expository in nature, we give a very detailed account of Fesenko’s work [1, 2, 3], thereby complementing those papers by including all the proofs. Let us describe how our paper is organized. In the first Section, we briefly review the Abelian local class field theory and the construction of the local Artin reciprocity map, following the Hazewinkel method and the Neukirch–Iwasawa method. -
Elliptic Curves of Odd Modular Degree
ISRAEL JOURNAL OF MATHEMATICS 169 (2009), 417–444 DOI: 10.1007/s11856-009-0017-x ELLIPTIC CURVES OF ODD MODULAR DEGREE BY Frank Calegari∗ and Matthew Emerton∗∗ Northwestern University, Department of Mathematics 2033 Sheridan Rd. Evanston, IL 60208 United States e-mail: [email protected], [email protected] ABSTRACT The modular degree mE of an elliptic curve E/Q is the minimal degree of any surjective morphism X0(N) E, where N is the conductor of E. → We give a necessary set of criteria for mE to be odd. In the case when N is prime our results imply a conjecture of Mark Watkins. As a technical tool, we prove a certain multiplicity one result at the prime p = 2, which may be of independent interest. 1. Introduction Let E be an elliptic curve over Q of conductor N. Since E is modular [3], there exists a surjective map π : X (N) E defined over Q. There is a unique such 0 → map of minimal degree (up to composing with automorphisms of E), and its degree mE is known as the modular degree of E. This degree has been much studied, both in relation to congruences between modular forms [38] and to the Selmer group of the symmetric square of E [14], [15], [35]. Since this Selmer ∗ Supported in part by the American Institute of Mathematics. ∗∗ Supported in part by NSF grant DMS-0401545. Receved August 11, 2006 and in revised form June 28, 2007 417 418 FRANKCALEGARIANDMATTHEWEMERTON Isr. J.Math. group can be considered as an elliptic analogue of the class group, one might expect in analogy with genus theory to find that mE satisfies certain divisibility properties, especially, perhaps, by the prime 2.