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8 Complete Fields and Valuation Rings
18.785 Number theory I Fall 2016 Lecture #8 10/04/2016 8 Complete fields and valuation rings In order to make further progress in our investigation of finite extensions L=K of the fraction field K of a Dedekind domain A, and in particular, to determine the primes p of K that ramify in L, we introduce a new tool that will allows us to \localize" fields. We have already seen how useful it can be to localize the ring A at a prime ideal p. This process transforms A into a discrete valuation ring Ap; the DVR Ap is a principal ideal domain and has exactly one nonzero prime ideal, which makes it much easier to study than A. By Proposition 2.7, the localizations of A at its prime ideals p collectively determine the ring A. Localizing A does not change its fraction field K. However, there is an operation we can perform on K that is analogous to localizing A: we can construct the completion of K with respect to one of its absolute values. When K is a global field, this process yields a local field (a term we will define in the next lecture), and we can recover essentially everything we might want to know about K by studying its completions. At first glance taking completions might seem to make things more complicated, but as with localization, it actually simplifies matters considerably. For those who have not seen this construction before, we briefly review some background material on completions, topological rings, and inverse limits. -
Determining Large Groups and Varieties from Small Subgroups and Subvarieties
Determining large groups and varieties from small subgroups and subvarieties This course will consist of two parts with the common theme of analyzing a geometric object via geometric sub-objects which are `small' in an appropriate sense. In the first half of the course, we will begin by describing the work on H. W. Lenstra, Jr., on finding generators for the unit groups of subrings of number fields. Lenstra discovered that from the point of view of computational complexity, it is advantageous to first consider the units of the ring of S-integers of L for some moderately large finite set of places S. This amounts to allowing denominators which only involve a prescribed finite set of primes. We will de- velop a generalization of Lenstra's method which applies to find generators of small height for the S-integral points of certain algebraic groups G defined over number fields. We will focus on G which are compact forms of GLd for d ≥ 1. In the second half of the course, we will turn to smooth projective surfaces defined by arithmetic lattices. These are Shimura varieties of complex dimension 2. We will try to generate subgroups of finite index in the fundamental groups of these surfaces by the fundamental groups of finite unions of totally geodesic projective curves on them, that is, via immersed Shimura curves. In both settings, the groups we will study are S-arithmetic lattices in a prod- uct of Lie groups over local fields. We will use the structure of our generating sets to consider several open problems about the geometry, group theory, and arithmetic of these lattices. -
Little Survey on Large Fields – Old & New –
Little survey on large fields { Old & New { Florian Pop ∗ Abstract. The large fields were introduced by the author in [59] and subsequently acquired several other names. This little survey includes earlier and new developments, and at the end of each section we mention a few open questions. 2010 Mathematics Subject Classification. Primary 12E, 12F, 12G, 12J. Secondary 12E30, 12F10, 12G99. Keywords. Large fields, ultraproducts, PAC, pseudo closed fields, Henselian pairs, elementary equivalence, algebraic varieties, rational points, function fields, (inverse) Galois theory, embedding problems, model theory, rational connectedness, extremal fields. Introduction The notion of large field was introduced in Pop [59] and proved to be the \right class" of fields over which one can do a lot of interesting mathematics, like (inverse) Galois theory, see Colliot-Th´el`ene[7], Moret-Bailly [45], Pop [59], [61], the survey article Harbater [30], study torsors of finite groups Moret- Bailly [46], study rationally connected varieties Koll´ar[37], study the elementary theory of function fields Koenigsmann [35], Poonen{Pop [55], characterize extremal valued fields as introduced by Ershov [12], see Azgin{Kuhlmann{Pop [1], etc. Maybe that is why the \large fields” acquired several other names |google it: ´epais,fertile, weite K¨orper, ample, anti-Mordellic. Last but not least, see Jarden's book [34] for more about large fields (which he calls \ample fields” in his book [34]), and Kuhlmann [41] for relations between large fields and local uniformization (`ala Zariski). Definition. A field k is called a large field, if for every irreducible k-curve C the following holds: If C has a k-rational smooth point, then C has infinitely many k-rational points. -
Local-Global Methods in Algebraic Number Theory
LOCAL-GLOBAL METHODS IN ALGEBRAIC NUMBER THEORY ZACHARY KIRSCHE Abstract. This paper seeks to develop the key ideas behind some local-global methods in algebraic number theory. To this end, we first develop the theory of local fields associated to an algebraic number field. We then describe the Hilbert reciprocity law and show how it can be used to develop a proof of the classical Hasse-Minkowski theorem about quadratic forms over algebraic number fields. We also discuss the ramification theory of places and develop the theory of quaternion algebras to show how local-global methods can also be applied in this case. Contents 1. Local fields 1 1.1. Absolute values and completions 2 1.2. Classifying absolute values 3 1.3. Global fields 4 2. The p-adic numbers 5 2.1. The Chevalley-Warning theorem 5 2.2. The p-adic integers 6 2.3. Hensel's lemma 7 3. The Hasse-Minkowski theorem 8 3.1. The Hilbert symbol 8 3.2. The Hasse-Minkowski theorem 9 3.3. Applications and further results 9 4. Other local-global principles 10 4.1. The ramification theory of places 10 4.2. Quaternion algebras 12 Acknowledgments 13 References 13 1. Local fields In this section, we will develop the theory of local fields. We will first introduce local fields in the special case of algebraic number fields. This special case will be the main focus of the remainder of the paper, though at the end of this section we will include some remarks about more general global fields and connections to algebraic geometry. -
Branching Laws for Classical Groups: the Non-Tempered Case
BRANCHING LAWS FOR CLASSICAL GROUPS: THE NON-TEMPERED CASE WEE TECK GAN, BENEDICT H. GROSS AND DIPENDRA PRASAD August 18, 2020 ABSTRACT. This paper generalizes the GGP conjectures which were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the nongeneric L-packets arising from Arthur parameters. The paper introduces the key no- tionof a relevant pair of A-parameters which governs the branching laws for GLn and all classical groups overboth local fields and global fields. It plays a role for all the branching problems studied in [GGP] including Bessel models and Fourier-Jacobi models. CONTENTS 1. Introduction 1 2. Notation and Preliminaries 8 3. Relevant Pair of A-Parameters 10 4. Correlator 12 5. Local Conjecture for GLn 15 6. Local Conjecture for Classical Groups 23 7. A Conjecture for A-packets 25 8. A Special Case of the Conjecture 31 9. Global Conjecture 34 10. Revisiting the Global Conjecture 43 11. Low Rank Examples 46 12. Automorphic Descent 55 13. L-functions: GL case 59 14. L-functions: Classical Groups 63 References 67 arXiv:1911.02783v2 [math.RT] 17 Aug 2020 1. INTRODUCTION This paper is a sequel to our earlier work [GP1, GP2, GGP, GGP2], which discussed several restriction (or branching) problems in the representation theory of classical groups. In the local case, the conjectural answer was given in terms of symplectic root numbers 1991 Mathematics Subject Classification. Primary 11F70; Secondary 22E55. WTG is partially supported by an MOE Tier 2 grant R146-000-233-112. DP thanks Science and En- gineering research board of the Department of Science and Technology, India for its support through the JC Bose National Fellowship of the Govt. -
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. -
Arithmetic Deformation Theory Via Arithmetic Fundamental Groups and Nonarchimedean Theta-Functions, Notes on the Work of Shinichi Mochizuki
ARITHMETIC DEFORMATION THEORY VIA ARITHMETIC FUNDAMENTAL GROUPS AND NONARCHIMEDEAN THETA-FUNCTIONS, NOTES ON THE WORK OF SHINICHI MOCHIZUKI IVAN FESENKO ABSTRACT. These notes survey the main ideas, concepts and objects of the work by Shinichi Mochizuki on inter- universal Teichmüller theory [31], which might also be called arithmetic deformation theory, and its application to diophantine geometry. They provide an external perspective which complements the review texts [32] and [33]. Some important developments which preceded [31] are presented in the first section. Several key aspects of arithmetic deformation theory are discussed in the second section. Its main theorem gives an inequality–bound on the size of volume deformation associated to a certain log-theta-lattice. The application to several fundamental conjectures in number theory follows from a further direct computation of the right hand side of the inequality. The third section considers additional related topics, including practical hints on how to study the theory. This text was published in Europ. J. Math. (2015) 1:405–440. CONTENTS Foreword 2 1. The origins 2 1.1. From class field theory to reconstructing number fields to coverings of P1 minus three points2 1.2. A development in diophantine geometry3 1.3. Conjectural inequalities for the same property4 1.4. A question posed to a student by his thesis advisor6 1.5. On anabelian geometry6 2. On arithmetic deformation theory8 2.1. Texts related to IUT8 2.2. Initial data 9 2.3. A brief outline of the proof and a list of some of the main concepts 10 2.4. Mono-anabelian geometry and multiradiality 12 2.5. -
On the Structure of Galois Groups As Galois Modules
ON THE STRUCTURE OF GALOIS GROUPS AS GALOIS MODULES Uwe Jannsen Fakult~t f~r Mathematik Universit~tsstr. 31, 8400 Regensburg Bundesrepublik Deutschland Classical class field theory tells us about the structure of the Galois groups of the abelian extensions of a global or local field. One obvious next step is to take a Galois extension K/k with Galois group G (to be thought of as given and known) and then to investigate the structure of the Galois groups of abelian extensions of K as G-modules. This has been done by several authors, mainly for tame extensions or p-extensions of local fields (see [10],[12],[3] and [13] for example and further literature) and for some infinite extensions of global fields, where the group algebra has some nice structure (Iwasawa theory). The aim of these notes is to show that one can get some results for arbitrary Galois groups by using the purely algebraic concept of class formations introduced by Tate. i. Relation modules. Given a presentation 1 + R ÷ F ÷ G~ 1 m m of a finite group G by a (discrete) free group F on m free generators, m the factor commutator group Rabm = Rm/[Rm'R m] becomes a finitely generated Z[G]-module via the conjugation in F . By Lyndon [19] and m Gruenberg [8]§2 we have 1.1. PROPOSITION. a) There is an exact sequence of ~[G]-modules (I) 0 ÷ R ab + ~[G] m ÷ I(G) + 0 , m where I(G) is the augmentation ideal, defined by the exact sequence (2) 0 + I(G) ÷ Z[G] aug> ~ ÷ 0, aug( ~ aoo) = ~ a . -
CHARACTERS of ALGEBRAIC GROUPS OVER NUMBER FIELDS Bachir Bekka, Camille Francini
CHARACTERS OF ALGEBRAIC GROUPS OVER NUMBER FIELDS Bachir Bekka, Camille Francini To cite this version: Bachir Bekka, Camille Francini. CHARACTERS OF ALGEBRAIC GROUPS OVER NUMBER FIELDS. 2020. hal-02483439 HAL Id: hal-02483439 https://hal.archives-ouvertes.fr/hal-02483439 Preprint submitted on 18 Feb 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. CHARACTERS OF ALGEBRAIC GROUPS OVER NUMBER FIELDS BACHIR BEKKA AND CAMILLE FRANCINI Abstract. Let k be a number field, G an algebraic group defined over k, and G(k) the group of k-rational points in G: We deter- mine the set of functions on G(k) which are of positive type and conjugation invariant, under the assumption that G(k) is gener- ated by its unipotent elements. An essential step in the proof is the classification of the G(k)-invariant ergodic probability measures on an adelic solenoid naturally associated to G(k); this last result is deduced from Ratner's measure rigidity theorem for homogeneous spaces of S-adic Lie groups. 1. Introduction Let k be a field and G an algebraic group defined over k. When k is a local field (that is, a non discrete locally compact field), the group G = G(k) of k-rational points in G is a locally compact group for the topology induced by k. -
Notes on Galois Cohomology—Modularity Seminar
Notes on Galois Cohomology—Modularity seminar Rebecca Bellovin 1 Introduction We’ve seen that the tangent space to a deformation functor is a Galois coho- mology group H1, and we’ll see that obstructions to a deformation problem will be in H2. So if we want to know things like the dimension of R or whether a deformation functor is smooth, we need to be able to get our hands on the cohomology groups. Secondarily, if we want to “deform sub- ject to conditions”, we’ll want to express the tangent space and obstruction space of those functors as cohomology groups, and cohomology groups we can compute in terms of an unrestricted deformation problem. For the most part, we will assume the contents of Serre’s Local Fields and Galois Cohomology. These cover the cases when G is finite (and discrete) and M is discrete, and G is profinite and M is discrete, respectively. References: Serre’s Galois Cohomology Neukirch’s Cohomology of Number Fields Appendix B of Rubin’s Euler Systems Washington’s article in CSS Darmon, Diamond, and Taylor (preprint on Darmon’s website) 2 Generalities Let G be a group, and let M be a module with an action by G. Both G and M have topologies; often both will be discrete (and G will be finite), 1 or G will be profinite with M discrete; or both will be profinite. We always require the action of G on M to be continuous. Let’s review group cohomology, using inhomogenous cocycles. For a topological group G and a topological G-module M, the ith group of continuous cochains Ci(G, M) is the group of continuous maps Gi → M. -
Non-Vanishing of L-Functions, the Ramanujan Conjecture, and Families of Hecke Characters
NON-VANISHING OF L-FUNCTIONS, THE RAMANUJAN CONJECTURE, AND FAMILIES OF HECKE CHARACTERS VALENTIN BLOMER AND FARRELL BRUMLEY Abstract. We prove a non-vanishing result for families of GLn × GLn Rankin-Selberg L-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on GLn. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group. 1. Introduction In this paper we prove a non-vanishing result for Rankin-Selberg L-functions for cusp forms on GLn when one factor ranges over twists by infinite order Hecke characters and give applications to bounds towards the Ramunujan conjecture. 1.1. Infinite order Hecke characters. Before stating our precise results, we enunciate two ways, one automorphic, the other arithmetic, in which infinite order Hecke characters have been put to use to solve problems in number theory whose formulation is naturally posed over number fields. The first is the principle that a cusp form (for the group GL2 at least) should always be taken together with its twists by mod 1 Grossencharacters. This philosophy has been attributed to Sarnak and has its origins in the hybrid mean value estimate of [31]. A striking justification is the recent article by Booker and Krishnamurthy [5] where it is shown that for an irreducible two dimensional complex representation ρ of the Weil-Deligne group, if the Artin-Weil L-function L(s; ρ !) is entire for all mod 1 Grossencharacters ! then ρ is automorphic. -
Zeros of Dirichlet L-Functions Over Function Fields
ZEROS OF DIRICHLET L-FUNCTIONS OVER FUNCTION FIELDS JULIO C. ANDRADE, STEVEN J. MILLER, KYLE PRATT, AND MINH-TAM TRINH ABSTRACT. Random matrix theory has successfully modeled many systems in physics and math- ematics, and often the analysis and results in one area guide development in the other. Hughes and Rudnick computed 1-level density statistics for low-lying zeros of the family of primitive Dirichlet L-functions of fixed prime conductor Q, as Q , and verified the unitary symmetry predicted by random matrix theory. We compute 1- and 2→-level ∞ statistics of the analogous family of Dirichlet L-functions over Fq(T ). Whereas the Hughes-Rudnick results were restricted by the support of the Fourier transform of their test function, our test function is periodic and our results are only restricted by a decay condition on its Fourier coefficients. We show the main terms agree with unitary sym- metry, and also isolate error terms. In concluding, we discuss an Fq(T )-analogue of Montgomery’s Hypothesis on the distribution of primes in arithmetic progressions, which Fiorilli and Miller show would remove the restriction on the Hughes-Rudnick results. CONTENTS 1. Introduction 1 2. Dirichlet L-Function Preliminaries 6 3. The 1-Level Global Regime 7 4. The 2-Level Global Regime 11 5. The Local Regime 13 6. Montgomery’s Hypothesis 16 Appendix A. Proof of the Explicit Formula 18 Appendix B. Traces of the Frobenius Class 19 References 20 1. INTRODUCTION arXiv:1404.2435v1 [math.NT] 9 Apr 2014 1.1. Background. In the 1970s, Montgomery and Dyson conjectured that local statistics of crit- ical zeros of the Riemann zeta function—in the limit of large height—should match those of angles of eigenvalues of matrices in the Gaussian Unitary Ensemble (GUE), which Wigner, Dyson and others (see [FirM] for a historical overview) had already used with great success in modeling the energy levels of heavy nuclei.