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Pila, O-Minimality and Diophantine Geometry
O-minimality and Diophantine geometry Jonathan Pila Abstract. This lecture is concerned with some recent applications of mathematical logic to Diophantine geometry. More precisely it concerns applications of o-minimality, a branch of model theory which treats tame structures in real geometry, to certain finiteness problems descending from the classical conjecture of Mordell. Mathematics Subject Classification (2010). Primary 03C64, 11G18. Keywords. O-minimal structure, Andr´e-Oortconjecture, Zilber-Pink conjecture. 1. Introduction This is a somewhat expanded version of my lecture at ICM 2014 in Seoul. It surveys some recent interactions between model theory and Diophantine geometry. The Diophantine problems to be considered are of a type descending from the classical Mordell conjecture (theorem of Faltings). I will describe the passage from Mordell's conjecture to the far-reaching Zilber-Pink conjecture, which is very much open and the subject of lively study by a variety of methods on several fronts. The model theory is \o-minimality", which studies tame structures in real geometry, and offers powerful tools applicable to certain “definable” sets. In combination with an elementary analytic method for \counting rational points" it leads to a general result about the height distribution of rational points on definable sets. This result can be successfully applied to Zilber-Pink problems in the presence of certain functional transcendence and arithmetic ingredients which are known in many cases but seemingly quite difficult in general. Both the methods and problems have connections with transcendental number theory. My further objective is to explain these connections and to bring out the pervasive presence of Schanuel's conjecture. -
Algebra & Number Theory Vol. 7 (2013)
Algebra & Number Theory Volume 7 2013 No. 3 msp Algebra & Number Theory msp.org/ant EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Raman Parimala Emory University, USA John H. Coates University of Cambridge, UK Jonathan Pila University of Oxford, UK J-L. Colliot-Thélène CNRS, Université Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hélène Esnault Freie Universität Berlin, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universität, Germany Joseph H. Silverman Brown University, USA Edward Frenkel University of California, Berkeley, USA Michael Singer North Carolina State University, USA Andrew Granville Université de Montréal, Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Virginia, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale University, USA Ravi Vakil Stanford University, -
On the Left Ideal in the Universal Enveloping
proceedings of the american mathematical society Volume 121, Number 4, August 1994 ON THE LEFT IDEAL IN THE UNIVERSALENVELOPING ALGEBRA OF A LIE GROUP GENERATED BY A COMPLEX LIE SUBALGEBRA JUAN TIRAO (Communicated by Roe W. Goodman) Abstract. Let Go be a connected Lie group with Lie algebra go and 'et n be a Lie subalgebra of the complexification g of go . Let C°°(Go)* be the annihilator of h in C°°(Go) and let s/ =^{Ccc(G0)h) be the annihilator of C00(Go)A in the universal enveloping algebra %(g) of g. If h is the complexification of the Lie algebra ho of a Lie subgroup Ho of Go then sf = %({g)h whenever Ho is closed, is a known result, and the point of this paper is to prove the converse assertion. The paper has two distinct parts, one for C°° , the other for holomorphic functions. In the first part the Lie algebra ho of the closure of Ha is characterized as the annihilator in go of C°°(Go)h , and it is proved that ho is an ideal in /i0 and that ho = ho © v where v is an abelian subalgebra of Ao . In the second part we consider a complexification G of Go and assume that h is the Lie algebra of a closed connected subgroup H of G. Then we establish that s/{cf(G)h) = %?(g)h if and only if G/H has many holomorphic functions. This is the case if G/H is a quasi-affine variety. From this we get that if H is a unipotent subgroup of G or if G and H are reductive groups then s/(C°°(G0)A) = iY(g)h . -
Lie Groups and Lie Algebras
2 LIE GROUPS AND LIE ALGEBRAS 2.1 Lie groups The most general definition of a Lie group G is, that it is a differentiable manifold with group structure, such that the multiplication map G G G, (g,h) gh, and the inversion map G G, g g−1, are differentiable.× → But we shall7→ not need this concept in full generality.→ 7→ In order to avoid more elaborate differential geometry, we will restrict attention to matrix groups. Consider the set of all invertible n n matrices with entries in F, where F stands either for the real (R) or complex× (C) numbers. It is easily verified to form a group which we denote by GL(n, F), called the general linear group in n dimensions over the number field F. The space of all n n matrices, including non- invertible ones, with entries in F is denoted by M(n,×F). It is an n2-dimensional 2 vector space over F, isomorphic to Fn . The determinant is a continuous function det : M(n, F) F and GL(n, F) = det−1(F 0 ), since a matrix is invertible → −{ } 2 iff its determinant is non zero. Hence GL(n, F) is an open subset of Fn . Group multiplication and inversion are just given by the corresponding familiar matrix 2 2 2 2 2 operations, which are differentiable functions Fn Fn Fn and Fn Fn respectively. × → → 2.1.1 Examples of Lie groups GL(n, F) is our main example of a (matrix) Lie group. Any Lie group that we encounter will be a subgroups of some GL(n, F). -
Introduction to Representations Theory of Lie Groups
Introduction to Representations Theory of Lie Groups Raul Gomez October 14, 2009 Introduction The purpose of this notes is twofold. The first goal is to give a quick answer to the question \What is representation theory about?" To answer this, we will show by examples what are the most important results of this theory, and the problems that it is trying to solve. To make the answer short we will not develop all the formal details of the theory and we will give preference to examples over proofs. Few results will be proved, and in the ones were a proof is given, we will skip the technical details. We hope that the examples and arguments presented here will be enough to give the reader and intuitive but concise idea of the covered material. The second goal of the notes is to be a guide to the reader interested in starting a serious study of representation theory. Sometimes, when starting the study of a new subject, it's hard to understand the underlying motivation of all the abstract definitions and technical lemmas. It's also hard to know what is the ultimate goal of the subject and to identify the important results in the sea of technical lemmas. We hope that after reading this notes, the interested reader could start a serious study of representation theory with a clear idea of its goals and philosophy. Lets talk now about the material covered on this notes. In the first section we will state the celebrated Peter-Weyl theorem, which can be considered as a generalization of the theory of Fourier analysis on the circle S1. -
Kac-Moody Algebras and Applications
Kac-Moody Algebras and Applications Owen Barrett December 24, 2014 Abstract This article is an introduction to the theory of Kac-Moody algebras: their genesis, their con- struction, basic theorems concerning them, and some of their applications. We first record some of the classical theory, since the Kac-Moody construction generalizes the theory of simple finite-dimensional Lie algebras in a closely analogous way. We then introduce the construction and properties of Kac-Moody algebras with an eye to drawing natural connec- tions to the classical theory. Last, we discuss some physical applications of Kac-Moody alge- bras,includingtheSugawaraandVirosorocosetconstructions,whicharebasictoconformal field theory. Contents 1 Introduction2 2 Finite-dimensional Lie algebras3 2.1 Nilpotency.....................................3 2.2 Solvability.....................................4 2.3 Semisimplicity...................................4 2.4 Root systems....................................5 2.4.1 Symmetries................................5 2.4.2 The Weyl group..............................6 2.4.3 Bases...................................6 2.4.4 The Cartan matrix.............................7 2.4.5 Irreducibility...............................7 2.4.6 Complex root systems...........................7 2.5 The structure of semisimple Lie algebras.....................7 2.5.1 Cartan subalgebras............................8 2.5.2 Decomposition of g ............................8 2.5.3 Existence and uniqueness.........................8 2.6 Linear representations of complex semisimple Lie algebras...........9 2.6.1 Weights and primitive elements.....................9 2.7 Irreducible modules with a highest weight....................9 2.8 Finite-dimensional modules............................ 10 3 Kac-Moody algebras 10 3.1 Basic definitions.................................. 10 3.1.1 Construction of the auxiliary Lie algebra................. 11 3.1.2 Construction of the Kac-Moody algebra................. 12 3.1.3 Root space of the Kac-Moody algebra g(A) .............. -
Ax-Schanuel and Strong Minimality for the $ J $-Function
AX-SCHANUEL AND STRONG MINIMALITY FOR THE j-FUNCTION VAHAGN ASLANYAN Abstract. Let K := (K;+, ·, D, 0, 1) be a differentially closed field of character- istic 0 with field of constants C. In the first part of the paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation E(x, y) and the geometry of the fibres Us := {y : E(s,y) ∧ y∈ / C} where s is a non-constant element. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of Us. Moreover, the induced structure on the Cartesian powers of Us is given by special subvarieties. In particular, since the j-function satisfies an Ax-Schanuel inequality of the required form (due to Pila and Tsimerman), applying our results to the j-function we recover a theorem of Freitag and Scanlon stating that the differential equation of j defines a strongly minimal set with trivial geometry. In the second part of the paper we study strongly minimal sets in the j-reducts of differentially closed fields. Let Ej (x, y) be the (two-variable) differential equation of the j-function. We prove a Zilber style classification result for strongly minimal sets in the reduct K := (K;+, ·, Ej ). More precisely, we show that in K all strongly minimal sets are geometrically trivial or non-orthogonal to C. Our proof is based on the Ax-Schanuel theorem and a matching Existential Closedness statement which asserts that systems of equations in terms of Ej have solutions in K unless having a solution contradicts Ax-Schanuel. -
Lie Groups and Lie Algebras
Lie groups and Lie algebras Jonny Evans March 10, 2016 1 Lecture 1: Introduction To the students, past, present and future, who have/are/will taken/taking/take this course and to those interested parties who just read the notes and gave me feedback: thank you for providing the background level of enthusiasm, pedantry and general con- fusion needed to force me to improve these notes. All comments and corrections are very welcome. 1.1 What is a representation? Definition 1.1. A representation ρ of a group G on an n-dimensional vector space V over a field k is an assignment of a k-linear map ρ(g): V ! V to each group element g such that • ρ(gh) = ρ(g)ρ(h) • ρ(1G) = 1V . Here are some equivalent definitions. A representation is... 1. ...an assignment of an n-by-n matrix ρ(g) to each group element g such that ρ(gh) = ρ(g)ρ(h) (i.e. the matrices multiply together like the group elements they represent). This is clearly the same as the previous definition once you have picked a basis of V to identify linear maps with matrices; 2. ...a homomorphism ρ: G ! GL(V ). The multiplicativity condition is now hiding in the definition of homomorphism, which requires ρ(gh) = ρ(g)ρ(h); 3. ...an action of G on V by linear maps. In other words an action ρ~: G × V ! V such that, for each g 2 G, the map ρ(g): V ! V defined by ρ(g)(v) =ρ ~(g; v) is linear. -
Effective Sato-Tate Conjecture for Abelian Varieties and Applications
EFFECTIVE SATO–TATE CONJECTURE FOR ABELIAN VARIETIES AND APPLICATIONS ALINA BUCUR, FRANCESC FITE,´ AND KIRAN S. KEDLAYA Abstract. From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato–Tate conjecture for an abelian variety A defined over a number field k with connected Sato–Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato–Tate measure that only depends on certain invariants of A. We discuss three applications of this conditional result. First, for an abelian variety defined over k, we consider a variant of Linnik’s problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius trace attain the integral part of the Hasse–Weil bound. Third, for a pair of abelian varieties A and A′ defined over k with no common factors up to k-isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces of A and A′ have opposite sign. Contents 1. Introduction 1 Notations and terminology. 4 Acknowledgements 4 2. Conjectural framework 5 2.1. Sato–Tate groups 5 2.2. Motivic L-functions 5 3. Effective Sato–Tate Conjecture 7 3.1. Lie group theory background 8 3.2. A multivariate Vinogradov function 9 3.3. -
O-Minimality and the André-Oort Conjecture for Cn
Annals of Mathematics 173 (2011), 1779{1840 doi: 10.4007/annals.2011.173.3.11 O-minimality and the Andr´e-Oortconjecture for Cn By Jonathan Pila Abstract We give an unconditional proof of the Andr´e-Oortconjecture for arbi- trary products of modular curves. We establish two generalizations. The first includes the Manin-Mumford conjecture for arbitrary products of el- liptic curves defined over Q as well as Lang's conjecture for torsion points in powers of the multiplicative group. The second includes the Manin- Mumford conjecture for abelian varieties defined over Q. Our approach uses the theory of o-minimal structures, a part of Model Theory, and follows a strategy proposed by Zannier and implemented in three recent papers: a new proof of the Manin-Mumford conjecture by Pila-Zannier; a proof of a special (but new) case of Pink's relative Manin-Mumford conjecture by Masser-Zannier; and new proofs of certain known results of Andr´e-Oort- Manin-Mumford type by Pila. 1. Introduction In this paper we give an unconditional proof of the Andr´e-Oortconjecture for arbitrary products of modular curves. Under the Generalized Riemann Hypothesis for imaginary quadratic fields this result is due to Edixhoven [32], [34]; for n = 2 it is an unconditional result of Andr´e[3]. Our approach uses the theory of o-minimal structures, a part of Model Theory. It leads naturally to a more general result that is an \Andr´e-Oort-Manin-Mumford-Lang"statement for varieties of the form ` X = Y1 × · · · × Yn × E1 × · · · × Em × G ; where n; m; ` are nonnegative integers, Y1 = Γ1nH;:::;Yn = ΓnnH are modular curves corresponding to the quotient of the upper half-plane H by congruence subgroups Γi of SL2(Z), E1;:::;Em are elliptic curves defined over Q, and G = Gm(C) is the multiplicative group of nonzero complex numbers. -
22 Representations of SL2
22 Representations of SL2 Finite dimensional complex representations of the following are much the same: SL2(R), sl2R, sl2C (as a complex Lie algebra), su2R, and SU2. This is because finite dimensional representations of a simply connected Lie group are in bi- jection with representations of the Lie algebra. Complex representations of a REAL Lie algebra L correspond to complex representations of its complexifica- tion L ⊗ C considered as a COMPLEX Lie algebra. Note that complex representations of a COMPLEX Lie algebra L⊗C are not the same as complex representations of the REAL Lie algebra L⊗C =∼ L+L. The representations of the real Lie algebra correspond roughly to (reps of L)⊗(reps of L). Strictly speaking, SL2(R) is not simply connected, which is not important for finite dimensional representations. 2 Set Ω = 2EF + 2F E + H ∈ U(sl2R). The main point is that Ω commutes with sl2R. You can check this by brute force: [H, Ω] = 2 ([H, E]F + E[H, F ]) + ··· | {z } 0 [E, Ω] = 2[E, E]F + 2E[F, E] + 2[E, F ]E + 2F [E, E] + [E, H]H + H[E, H] = 0 [F, Ω] = Similar Thus, Ω is in the center of U(sl2R). In fact, it generates the center. This does not really explain where Ω comes from. Why does Ω exist? The answer is that it comes from a symmetric invariant bilinear form on the Lie algebra sl2R given by (E, F ) = 1, (E, E) = (F, F ) = (F, H) = (E, H) = 0, (H, H) = 2. This bilinear form is an invariant map L ⊗ L → C, where L = sl2R, which by duality gives an invariant element in L ⊗ L, which turns out to be 2E ⊗ F + 2F ⊗ E + H ⊗ H. -
2018-06-108.Pdf
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Feature S E European Tensor Product and Semi-Stability M M Mathematical Interviews E S Society Peter Sarnak Gigliola Staffilani June 2018 Obituary Robert A. Minlos Issue 108 ISSN 1027-488X Prague, venue of the EMS Council Meeting, 23–24 June 2018 New books published by the Individual members of the EMS, member S societies or societies with a reciprocity agree- E European ment (such as the American, Australian and M M Mathematical Canadian Mathematical Societies) are entitled to a discount of 20% on any book purchases, if E S Society ordered directly at the EMS Publishing House. Bogdan Nica (McGill University, Montreal, Canada) A Brief Introduction to Spectral Graph Theory (EMS Textbooks in Mathematics) ISBN 978-3-03719-188-0. 2018. 168 pages. Hardcover. 16.5 x 23.5 cm. 38.00 Euro Spectral graph theory starts by associating matrices to graphs – notably, the adjacency matrix and the Laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph theoretic or combinatorial. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context of the second, spectral, half. The text is enriched by many exercises and their solutions.