The Langlands Program
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Robert P. Langlands
The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2018 to Robert P. Langlands of the Institute for Advanced Study, Princeton, USA, “for his visionary program connecting representation theory to number theory.” The Langlands program predicts the existence of a tight The group GL(2) is the simplest example of a non- web of connections between automorphic forms and abelian reductive group. To proceed to the general Galois groups. case, Langlands saw the need for a stable trace formula, now established by Arthur. Together with Ngô’s proof The great achievement of algebraic number theory in of the so-called Fundamental Lemma, conjectured by the first third of the 20th century was class field theory. Langlands, this has led to the endoscopic classification This theory is a vast generalisation of Gauss’s law of of automorphic representations of classical groups, in quadratic reciprocity. It provides an array of powerful terms of those of general linear groups. tools for studying problems governed by abelian Galois groups. The non-abelian case turns out to be Functoriality dramatically unifies a number of important substantially deeper. Langlands, in a famous letter to results, including the modularity of elliptic curves and André Weil in 1967, outlined a far-reaching program that the proof of the Sato-Tate conjecture. It also lends revolutionised the understanding of this problem. weight to many outstanding conjectures, such as the Ramanujan-Peterson and Selberg conjectures, and the Langlands’s recognition that one should relate Hasse-Weil conjecture for zeta functions. representations of Galois groups to automorphic forms involves an unexpected and fundamental insight, now Functoriality for reductive groups over number fields called Langlands functoriality. -
Class Numbers of CM Algebraic Tori, CM Abelian Varieties and Components of Unitary Shimura Varieties
CLASS NUMBERS OF CM ALGEBRAIC TORI, CM ABELIAN VARIETIES AND COMPONENTS OF UNITARY SHIMURA VARIETIES JIA-WEI GUO, NAI-HENG SHEU, AND CHIA-FU YU Abstract. We give a formula for the class number of an arbitrary CM algebraic torus over Q. This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result. 1. Introduction An algebraic torus T over a number field k is a connected linear algebraic group over k such d that T k k¯ isomorphic to (Gm) k k¯ over the algebraic closure k¯ of k for some integer d 1. The class⊗ number, h(T ), of T is by⊗ definition, the cardinality of T (k) T (A )/U , where A≥ \ k,f T k,f is the finite adele ring of k and UT is the maximal open compact subgroup of T (Ak,f ). Asa natural generalization for the class number of a number field, Takashi Ono [18, 19] studied the class numbers of algebraic tori. Let K/k be a finite extension and let RK/k denote the Weil restriction of scalars form K to k, then we have the following exact sequence of tori defined over k 1 R(1) (G ) R (G ) G 1, −→ K/k m,K −→ K/k m,K −→ m,k −→ where R(1) (G ) is the kernel of the norm map N : R (G ) G . -
An Introduction to Iwasawa Theory
An Introduction to Iwasawa Theory Notes by Jim L. Brown 2 Contents 1 Introduction 5 2 Background Material 9 2.1 AlgebraicNumberTheory . 9 2.2 CyclotomicFields........................... 11 2.3 InfiniteGaloisTheory . 16 2.4 ClassFieldTheory .......................... 18 2.4.1 GlobalClassFieldTheory(ideals) . 18 2.4.2 LocalClassFieldTheory . 21 2.4.3 GlobalClassFieldTheory(ideles) . 22 3 SomeResultsontheSizesofClassGroups 25 3.1 Characters............................... 25 3.2 L-functionsandClassNumbers . 29 3.3 p-adicL-functions .......................... 31 3.4 p-adic L-functionsandClassNumbers . 34 3.5 Herbrand’sTheorem . .. .. .. .. .. .. .. .. .. .. 36 4 Zp-extensions 41 4.1 Introduction.............................. 41 4.2 PowerSeriesRings .......................... 42 4.3 A Structure Theorem on ΛO-modules ............... 45 4.4 ProofofIwasawa’stheorem . 48 5 The Iwasawa Main Conjecture 61 5.1 Introduction.............................. 61 5.2 TheMainConjectureandClassGroups . 65 5.3 ClassicalModularForms. 68 5.4 ConversetoHerbrand’sTheorem . 76 5.5 Λ-adicModularForms . 81 5.6 ProofoftheMainConjecture(outline) . 85 3 4 CONTENTS Chapter 1 Introduction These notes are the course notes from a topics course in Iwasawa theory taught at the Ohio State University autumn term of 2006. They are an amalgamation of results found elsewhere with the main two sources being [Wash] and [Skinner]. The early chapters are taken virtually directly from [Wash] with my contribution being the choice of ordering as well as adding details to some arguments. Any mistakes in the notes are mine. There are undoubtably type-o’s (and possibly mathematical errors), please send any corrections to [email protected]. As these are course notes, several proofs are omitted and left for the reader to read on his/her own time. -
Automorphic Functions and Fermat's Last Theorem(3)
Jiang and Wiles Proofs on Fermat Last Theorem (3) Abstract D.Zagier(1984) and K.Inkeri(1990) said[7]:Jiang mathematics is true,but Jiang determinates the irrational numbers to be very difficult for prime exponent p.In 1991 Jiang studies the composite exponents n=15,21,33,…,3p and proves Fermat last theorem for prime exponenet p>3[1].In 1986 Gerhard Frey places Fermat last theorem at the elliptic curve that is Frey curve.Andrew Wiles studies Frey curve. In 1994 Wiles proves Fermat last theorem[9,10]. Conclusion:Jiang proof(1991) is direct and simple,but Wiles proof(1994) is indirect and complex.If China mathematicians had supported and recognized Jiang proof on Fermat last theorem,Wiles would not have proved Fermat last theorem,because in 1991 Jiang had proved Fermat last theorem.Wiles has received many prizes and awards,he should thank China mathematicians. - Automorphic Functions And Fermat’s Last Theorem(3) (Fermat’s Proof of FLT) 1 Chun-Xuan Jiang P. O. Box 3924, Beijing 100854, P. R. China [email protected] Abstract In 1637 Fermat wrote: “It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.” This means: xyznnnn+ =>(2) has no integer solutions, all different from 0(i.e., it has only the trivial solution, where one of the integers is equal to 0). It has been called Fermat’s last theorem (FLT). -
Wei Zhang – Positivity of L-Functions and “Completion of the Square”
Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Positivity of L-functions and “Completion of square" Wei Zhang Massachusetts Institute of Technology Bristol, June 4th, 2018 1 Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Outline 1 Riemann hypothesis 2 Positivity of L-functions 3 Completion of square 4 Positivity on surfaces 2 Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Riemann Hypothesis (RH) Riemann zeta function 1 X 1 1 1 ζ(s) = = 1 + + + ··· ns 2s 3s n=1 Y 1 = ; s 2 ; Re(s) > 1 1 − p−s C p; primes Analytic continuation and Functional equation −s=2 Λ(s) := π Γ(s=2)ζ(s) = Λ(1 − s); s 2 C Conjecture The non-trivial zeros of the Riemann zeta function ζ(s) lie on the line 1 Re(s) = : 3 2 Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Riemann Hypothesis (RH) Riemann zeta function 1 X 1 1 1 ζ(s) = = 1 + + + ··· ns 2s 3s n=1 Y 1 = ; s 2 ; Re(s) > 1 1 − p−s C p; primes Analytic continuation and Functional equation −s=2 Λ(s) := π Γ(s=2)ζ(s) = Λ(1 − s); s 2 C Conjecture The non-trivial zeros of the Riemann zeta function ζ(s) lie on the line 1 Re(s) = : 3 2 Riemann hypothesis Positivity of L-functions Completion of square Positivity on surfaces Equivalent statements of RH Let π(X) = #fprimes numbers p ≤ Xg: Then Z X dt 1=2+ RH () π(X) − = O(X ): 2 log t Let X θ(X) = log p: p<X Then RH () jθ(X) − Xj = O(X 1=2+): 4 Riemann hypothesis Positivity of L-functions Completion of square -
Shtukas for Reductive Groups and Langlands Correspondence for Functions Fields
SHTUKAS FOR REDUCTIVE GROUPS AND LANGLANDS CORRESPONDENCE FOR FUNCTIONS FIELDS VINCENT LAFFORGUE This text gives an introduction to the Langlands correspondence for function fields and in particular to some recent works in this subject. We begin with a short historical account (all notions used below are recalled in the text). The Langlands correspondence [49] is a conjecture of utmost impor- tance, concerning global fields, i.e. number fields and function fields. Many excellent surveys are available, for example [39, 14, 13, 79, 31, 5]. The Langlands correspondence belongs to a huge system of conjectures (Langlands functoriality, Grothendieck’s vision of motives, special val- ues of L-functions, Ramanujan-Petersson conjecture, generalized Rie- mann hypothesis). This system has a remarkable deepness and logical coherence and many cases of these conjectures have already been es- tablished. Moreover the Langlands correspondence over function fields admits a geometrization, the “geometric Langlands program”, which is related to conformal field theory in Theoretical Physics. Let G be a connected reductive group over a global field F . For the sake of simplicity we assume G is split. The Langlands correspondence relates two fundamental objects, of very different nature, whose definition will be recalled later, • the automorphic forms for G, • the global Langlands parameters , i.e. the conjugacy classes of morphisms from the Galois group Gal(F =F ) to the Langlands b dual group G(Q`). b For G = GL1 we have G = GL1 and this is class field theory, which describes the abelianization of Gal(F =F ) (one particular case of it for Q is the law of quadratic reciprocity, which dates back to Euler, Legendre and Gauss). -
RELATIVE LANGLANDS This Is Based on Joint Work with Yiannis
RELATIVE LANGLANDS DAVID BEN-ZVI This is based on joint work with Yiannis Sakellaridis and Akshay Venkatesh. The general plan is to explain a connection between physics and number theory which goes through the intermediary: extended topological field theory (TFT). The moral is that boundary conditions for N = 4 super Yang-Mills (SYM) lead to something about periods of automorphic forms. Slogan: the relative Langlands program can be explained via relative TFT. 1. Periods of automorphic forms on H First we provide some background from number theory. Recall we can picture the upper-half-space H as in fig. 1. We are thinking of a modular form ' as a holomorphic function on H which transforms under the modular group SL2 (Z), or in general some congruent sub- group Γ ⊂ SL2 (Z), like a k=2-form (differential form) and is holomorphic at 1. We will consider some natural measurements of '. In particular, we can \measure it" on the red and blue lines in fig. 1. Note that we can also think of H as in fig. 2, where the red and blue lines are drawn as well. Since ' is invariant under SL2 (Z), it is really a periodic function on the circle, so it has a Fourier series. The niceness at 1 condition tells us that it starts at 0, so we get: X n (1) ' = anq n≥0 Date: Tuesday March 24, 2020; Thursday March 26, 2020. Notes by: Jackson Van Dyke, all errors introduced are my own. Figure 1. Fundamental domain for the action of SL2 (Z) on H in gray. -
Introduction to L-Functions: Dedekind Zeta Functions
Introduction to L-functions: Dedekind zeta functions Paul Voutier CIMPA-ICTP Research School, Nesin Mathematics Village June 2017 Dedekind zeta function Definition Let K be a number field. We define for Re(s) > 1 the Dedekind zeta function ζK (s) of K by the formula X −s ζK (s) = NK=Q(a) ; a where the sum is over all non-zero integral ideals, a, of OK . Euler product exists: Y −s −1 ζK (s) = 1 − NK=Q(p) ; p where the product extends over all prime ideals, p, of OK . Re(s) > 1 Proposition For any s = σ + it 2 C with σ > 1, ζK (s) converges absolutely. Proof: −n Y −s −1 Y 1 jζ (s)j = 1 − N (p) ≤ 1 − = ζ(σ)n; K K=Q pσ p p since there are at most n = [K : Q] many primes p lying above each rational prime p and NK=Q(p) ≥ p. A reminder of some algebraic number theory If [K : Q] = n, we have n embeddings of K into C. r1 embeddings into R and 2r2 embeddings into C, where n = r1 + 2r2. We will label these σ1; : : : ; σr1 ; σr1+1; σr1+1; : : : ; σr1+r2 ; σr1+r2 . If α1; : : : ; αn is a basis of OK , then 2 dK = (det (σi (αj ))) : Units in OK form a finitely-generated group of rank r = r1 + r2 − 1. Let u1;:::; ur be a set of generators. For any embedding σi , set Ni = 1 if it is real, and Ni = 2 if it is complex. Then RK = det (Ni log jσi (uj )j)1≤i;j≤r : wK is the number of roots of unity contained in K. -
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. -
Notes on the Riemann Hypothesis Ricardo Pérez-Marco
Notes on the Riemann Hypothesis Ricardo Pérez-Marco To cite this version: Ricardo Pérez-Marco. Notes on the Riemann Hypothesis. 2018. hal-01713875 HAL Id: hal-01713875 https://hal.archives-ouvertes.fr/hal-01713875 Preprint submitted on 21 Feb 2018 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. NOTES ON THE RIEMANN HYPOTHESIS RICARDO PEREZ-MARCO´ Abstract. Our aim is to give an introduction to the Riemann Hypothesis and a panoramic view of the world of zeta and L-functions. We first review Riemann's foundational article and discuss the mathematical background of the time and his possible motivations for making his famous conjecture. We discuss some of the most relevant developments after Riemann that have contributed to a better understanding of the conjecture. Contents 1. Euler transalgebraic world. 2 2. Riemann's article. 8 2.1. Meromorphic extension. 8 2.2. Value at negative integers. 10 2.3. First proof of the functional equation. 11 2.4. Second proof of the functional equation. 12 2.5. The Riemann Hypothesis. 13 2.6. The Law of Prime Numbers. 17 3. On Riemann zeta-function after Riemann. -
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. -
Of Triangles, Gas, Price, and Men
OF TRIANGLES, GAS, PRICE, AND MEN Cédric Villani Univ. de Lyon & Institut Henri Poincaré « Mathematics in a complex world » Milano, March 1, 2013 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemannian (= non-Euclidean) geometry At each location, the units of length and angles may change Shortest path (= geodesics) are curved!! Geodesics can tend to get closer (positive curvature, fat triangles) or to get further apart (negative curvature, skinny triangles) Hyperbolic surfaces Bernhard Riemann 1826-1866 List of topics named after Bernhard Riemann From Wikipedia, the free encyclopedia Riemann singularity theorem Cauchy–Riemann equations Riemann solver Compact Riemann surface Riemann sphere Free Riemann gas Riemann–Stieltjes integral Generalized Riemann hypothesis Riemann sum Generalized Riemann integral Riemann surface Grand Riemann hypothesis Riemann theta function Riemann bilinear relations Riemann–von Mangoldt formula Riemann–Cartan geometry Riemann Xi function Riemann conditions Riemann zeta function Riemann curvature tensor Zariski–Riemann space Riemann form Riemannian bundle metric Riemann function Riemannian circle Riemann–Hilbert correspondence Riemannian cobordism Riemann–Hilbert problem Riemannian connection Riemann–Hurwitz formula Riemannian cubic polynomials Riemann hypothesis Riemannian foliation Riemann hypothesis for finite fields Riemannian geometry Riemann integral Riemannian graph Bernhard