Current Trends and Open Problems in Arithmetic Dynamics
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CHAPTER 0 PRELIMINARY MATERIAL Paul
CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic preliminary notation, both mathematical and logistical. Sec- tion 2 describes what algebraic geometry is assumed of the reader, and gives a few conventions that will be assumed here. Section 3 gives a few more details on the field of definition of a variety. Section 4 does the same as Section 2 for number theory. The remaining sections of this chapter give slightly longer descriptions of some topics in algebraic geometry that will be needed: Kodaira’s lemma in Section 5, and descent in Section 6. x1. General notation The symbols Z , Q , R , and C stand for the ring of rational integers and the fields of rational numbers, real numbers, and complex numbers, respectively. The symbol N sig- nifies the natural numbers, which in this book start at zero: N = f0; 1; 2; 3;::: g . When it is necessary to refer to the positive integers, we use subscripts: Z>0 = f1; 2; 3;::: g . Similarly, R stands for the set of nonnegative real numbers, etc. ¸0 ` ` The set of extended real numbers is the set R := f¡1g R f1g . It carries the obvious ordering. ¯ ¯ If k is a field, then k denotes an algebraic closure of k . If ® 2 k , then Irr®;k(X) is the (unique) monic irreducible polynomial f 2 k[X] for which f(®) = 0 . Unless otherwise specified, the wording almost all will mean all but finitely many. -
Discrete and Profinite Groups Acting on Regular Rooted Trees
Discrete and Profinite Groups Acting on Regular Rooted Trees Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades “Doctor rerum naturalium” der Georg-August-Universit¨at G¨ottingen vorgelegt von Olivier Siegenthaler aus Lausanne G¨ottingen, den 31. August 2009 Referent: Prof. Dr. Laurent Bartholdi Korreferent: Prof. Dr. Thomas Schick Tag der m¨undlichen Pr¨ufung: den 28. September 2009 Contents Introduction 1 1 Foundations 5 1.1 Definition of Aut X∗ and ...................... 5 A∗ 1.2 ZariskiTopology ............................ 7 1.3 Actions of X∗ .............................. 8 1.4 Self-SimilarityandBranching . 9 1.5 Decompositions and Generators of Aut X∗ and ......... 11 A∗ 1.6 The Permutation Modules k X and k X ............ 13 { } {{ }} 1.7 Subgroups of Aut X∗ .......................... 14 1.8 RegularBranchGroups . 16 1.9 Self-Similarity and Branching Simultaneously . 18 1.10Questions ................................ 19 ∗ 2 The Special Case Autp X 23 2.1 Definition ................................ 23 ∗ 2.2 Subgroups of Autp X ......................... 24 2.3 NiceGeneratingSets. 26 2.4 Uniseriality ............................... 28 2.5 Signature and Maximal Subgroups . 30 2.6 Torsion-FreeGroups . 31 2.7 Some Specific Classes of Automorphisms . 32 2.8 TorsionGroups ............................. 34 3 Wreath Product of Affine Group Schemes 37 3.1 AffineSchemes ............................. 38 3.2 ExponentialObjects . 40 3.3 Some Hopf Algebra Constructions . 43 3.4 Group Schemes Corresponding to Aut X∗ .............. 45 3.5 Iterated Wreath Product of the Frobenius Kernel . 46 i Contents 4 Central Series and Automorphism Towers 49 4.1 Notation................................. 49 4.2 CentralSeries.............................. 51 4.3 Automorphism and Normalizer Towers . 54 5 Hausdorff Dimension 57 5.1 Definition ................................ 57 5.2 Layers .................................. 58 5.3 ComputingDimensions . -
Abstract Quotients of Profinite Groups, After Nikolov and Segal
ABSTRACT QUOTIENTS OF PROFINITE GROUPS, AFTER NIKOLOV AND SEGAL BENJAMIN KLOPSCH Abstract. In this expanded account of a talk given at the Oberwolfach Ar- beitsgemeinschaft “Totally Disconnected Groups”, October 2014, we discuss results of Nikolay Nikolov and Dan Segal on abstract quotients of compact Hausdorff topological groups, paying special attention to the class of finitely generated profinite groups. Our primary source is [17]. Sidestepping all difficult and technical proofs, we present a selection of accessible arguments to illuminate key ideas in the subject. 1. Introduction §1.1. Many concepts and techniques in the theory of finite groups depend in- trinsically on the assumption that the groups considered are a priori finite. The theoretical framework based on such methods has led to marvellous achievements, including – as a particular highlight – the classification of all finite simple groups. Notwithstanding, the same methods are only of limited use in the study of infinite groups: it remains mysterious how one could possibly pin down the structure of a general infinite group in a systematic way. Significantly more can be said if such a group comes equipped with additional information, such as a structure-preserving action on a notable geometric object. A coherent approach to studying restricted classes of infinite groups is found by imposing suitable ‘finiteness conditions’, i.e., conditions that generalise the notion of being finite but are significantly more flexible, such as the group being finitely generated or compact with respect to a natural topology. One rather fruitful theme, fusing methods from finite and infinite group theory, consists in studying the interplay between an infinite group Γ and the collection of all its finite quotients. -
ON the FIELD of DEFINITION of P-TORSION POINTS on ELLIPTIC CURVES OVER the RATIONALS
ON THE FIELD OF DEFINITION OF p-TORSION POINTS ON ELLIPTIC CURVES OVER THE RATIONALS ALVARO´ LOZANO-ROBLEDO Abstract. Let SQ(d) be the set of primes p for which there exists a number field K of degree ≤ d and an elliptic curve E=Q, such that the order of the torsion subgroup of E(K) is divisible by p. In this article we give bounds for the primes in the set SQ(d). In particular, we show that, if p ≥ 11, p 6= 13; 37, and p 2 SQ(d), then p ≤ 2d + 1. Moreover, we determine SQ(d) for all d ≤ 42, and give a conjectural formula for all d ≥ 1. If Serre’s uniformity problem is answered positively, then our conjectural formula is valid for all sufficiently large d. Under further assumptions on the non-cuspidal points on modular curves that parametrize those j-invariants associated to Cartan subgroups, the formula is valid for all d ≥ 1. 1. Introduction Let K be a number field of degree d ≥ 1 and let E=K be an elliptic curve. The Mordell-Weil theorem states that E(K), the set of K-rational points on E, can be given the structure of a finitely ∼ R generated abelian group. Thus, there is an integer R ≥ 0 such that E(K) = E(K)tors ⊕ Z and the torsion subgroup E(K)tors is finite. Here, we will focus on the order of E(K)tors. In particular, we are interested in the following question: if we fix d ≥ 1, what are the possible prime divisors of the order of E(K)tors, for E and K as above? Definition 1.1. -
ARTIN-MAZUR ZETA FUNCTIONS in ARITHMETIC DYNAMICS How Should We Define the Zeta Function of a Rational Map F : P 1(K) → P 1(K)
ARTIN-MAZUR ZETA FUNCTIONS IN ARITHMETIC DYNAMICS EVAN WARNER How should we define the zeta function of a rational map f : P1(k) ! P1(k), k a field? All we need is a sequence of numbers, and it makes sense to consider the sequence jPern(f)j, where 1 n Pern(f) = fx 2 P (k): f (x) = xg: Then we construct a zeta function in the usual way: 1 ! X tn ζ(f; t) = exp jPer (f)j : n n n=1 This is a special case of an Artin-Mazur zeta function, which is defined for certain dynamical systems (and in general counts the number of isolated fixed points). Note that we are, as usual, not counting fixed points with multiplicity, which in this case would be something like the Lefschetz index. In fact, if we did count with n multiplicity and deg f = d, then jPern(f)j = d + 1 for all n, and after summing the geometric series we would get 1 ζ(f; t) = : (1 − t)(1 − dt) n In general, we at least have jPern(f)j ≤ d + 1, so the zeta function is certainly defined as a formal power series and even converges on a positive radius around the origin if k is a normed field. The usual formal manipulations give rise to an Euler product over cycles, and in particular ζ(f; t) 2 Z[[t]]. Over C, these zeta functions aren't too hard to calculate: Theorem 1 (Hinkkanen). If f : P1(C) ! P1(C) has degree d ≥ 2, then ζ(f; t) is a rational function. -
Burnside, March 2016
A Straightforward Solution to Burnside’s Problem S. Bachmuth 1. Introduction The Burnside Problem for groups asks whether a finitely generated group, all of whose elements have bounded order, is finite. We present a straightforward proof showing that the 2-generator Burnside groups of prime power exponent are solvable and therefore finite. This proof is straightforward in that it does not rely on induced maps as in [2], but it is strongly dependent on Theorem B in the joint paper with H. A. Heilbronn and H. Y. Mochizuki [9]. Theorem B is reformulated here as Lemma 3(i) in Section 2. Our use of Lemma 3(i) is indispensable. Throughout this paper we fix a prime power q = pe and unless specifically mentioned otherwise, all groups are 2-generator. At appropriate places, we may require e = 1 so that q = p is prime; otherwise q may be any fixed prime power. The only (published) positive results of finiteness of Burnside groups of prime power exponents are for exponents q = 2, 3 and 4 ([10],[12]). Some authors, beginning with P. S. Novikoff and S. I. Adian {15], (see also [1}), have claimed that groups of exponent k are infinite for k sufficiently large. Our result here, as in [2], is at odds with this claim. Since this proof avoids the use of induced maps, Section 4 of [2] has been rewritten. Sections 2, 3 and 6 have been left unaltered apart from minor, mostly expository, changes and renumbering of items. The introduction and Section 5 have been rewritten. Sections 5 & 6 are not involved in the proof although Section 5 is strongly recommended. -
FINITENESS of COMMUTABLE MAPS of BOUNDED DEGREE 1. Introduction a Dynamical System (V,F) Consists of a Set V and a Self Map
Bull. Korean Math. Soc. 52 (2015), No. 1, pp. 45–56 http://dx.doi.org/10.4134/BKMS.2015.52.1.045 FINITENESS OF COMMUTABLE MAPS OF BOUNDED DEGREE Chong Gyu Lee and Hexi Ye Abstract. In this paper, we study the relation between two dynamical systems (V,f) and (V,g) with f◦g = g◦f. As an application, we show that an endomorphism (respectively a polynomial map with Zariski dense, of bounded Preper(f)) has only finitely many endomorphisms (respectively polynomial maps) of bounded degree which are commutable with f. 1. Introduction A dynamical system (V,f) consists of a set V and a self map f : V → V . If V is a subset of a projective space Pn defined over a finitely generated field K over Q, then we have arithmetic height functions on V , which make a huge contribution to the study of (V,f). In this paper, we show the following results by studying arithmetic relations between two dynamical systems (V,f) and (V,g) with f ◦ g = g ◦ f. n n Main Theorem (Theorems 3.3 and 4.2). (1) Let φ : PC → PC be an en- n domorphism on PC, of degree at least 2. Then there are only finitely many endomorphisms of degree d which are commutable with φ: n Com(φ, d) := {ψ ∈ End(PC) | φ ◦ ψ = ψ ◦ φ, deg ψ = d} is a finite set. n n (2) Let f : AC → AC be a quasi-projective endomorphism (i.e., polyno- n mial maps) on AC. Suppose that Preper(f), the set of preperiodic points of f, is of bounded height and Zariski dense. -
Graduate Texts in Mathematics
Graduate Texts in Mathematics Editorial Board S. Axler F.W. Gehring K.A. Ribet BOOKS OF RELATED INTEREST BY SERGE LANG Math Talks for Undergraduates 1999, ISBN 0-387-98749-5 Linear Algebra, Third Edition 1987, ISBN 0-387-96412-6 Undergraduate Algebra, Second Edition 1990, ISBN 0-387-97279-X Undergraduate Analysis, Second Edition 1997, ISBN 0-387-94841-4 Complex Analysis, Third Edition 1993, ISBN 0-387-97886 Real and Functional Analysis, Third Edition 1993, ISBN 0-387-94001-4 Algebraic Number Theory, Second Edition 1994, ISBN 0-387-94225-4 OTHER BOOKS BY LANG PUBLISHED BY SPRINGER-VERLAG Introduction to Arakelov Theory • Riemann-Roch Algebra (with William Fulton) • Complex Multiplication • Introduction to Modular Forms • Modular Units (with Daniel Kubert) • Fundamentals of Diophantine Geometry • Elliptic Functions • Number Theory III • Survey of Diophantine Geometry • Fundamentals of Differential Geometry • Cyclotomic Fields I and II • SL2(R) • Abelian Varieties • Introduction to Algebraic and Abelian Functions • Introduction to Diophantine Approximations • Elliptic Curves: Diophantine Analysis • Introduction to Linear Algebra • Calculus of Several Variables • First Course in Calculus • Basic Mathematics • Geometry: A High School Course (with Gene Murrow) • Math! Encounters with High School Students • The Beauty of Doing Mathematics • THE FILE • CHALLENGES Serge Lang Algebra Revised Third Edition Springer Serge Lang Department of Mathematics Yale University New Haven, CT 96520 USA Editorial Board S. Axler Mathematics Department F.W. Gehring K.A. Ribet San Francisco State Mathematics Department Mathematics Department University East Hall University of California, San Francisco, CA 94132 University of Michigan Berkeley USA Ann Arbor, MI 48109 Berkeley, CA 94720-3840 [email protected] USA USA [email protected]. -
THE GALOIS ACTION on DESSIN D'enfants Contents 1. Introduction 1 2. Background: the Absolute Galois Group 4 3. Background
THE GALOIS ACTION ON DESSIN D'ENFANTS CARSON COLLINS Abstract. We introduce the theory of dessin d'enfants, with emphasis on how it relates to the absolute Galois group of Q. We prove Belyi's Theorem and show how the resulting Galois action is faithful on dessins. We also discuss the action on categories equivalent to dessins and prove its most powerful invariants for classifying orbits of the action. Minimal background in Galois theory or algebraic geometry is assumed, and we review those concepts which are necessary to this task. Contents 1. Introduction 1 2. Background: The Absolute Galois Group 4 3. Background: Algebraic Curves 4 4. Belyi's Theorem 5 5. The Galois Action on Dessins 6 6. Computing the Belyi Map of a Spherical Dessin 7 7. Faithfulness of the Galois Action 9 8. Cartographic and Automorphism Groups of a Dessin 10 9. Regular Dessins: A Galois Correspondence 13 Acknowledgments 16 References 16 1. Introduction Definition 1.1. A bigraph is a connected bipartite graph with a fixed 2-coloring into nonempty sets of black and white vertices. We refer to the edges of a bigraph as its darts. Definition 1.2. A dessin is a bigraph with an embedding into a topological surface such that its complement in the surface is a disjoint union of open disks, called the faces of the dessin. Dessin d'enfant, often abbreviated to dessin, is French for "child's drawing." As the definition above demonstrates, these are simple combinatorial objects, yet they have subtle and deep connections to algebraic geometry and Galois theory. -
Congruence Subgroup Problem for Algebraic Groups: Old and New Astérisque, Tome 209 (1992), P
Astérisque A. S. RAPINCHUK Congruence subgroup problem for algebraic groups: old and new Astérisque, tome 209 (1992), p. 73-84 <http://www.numdam.org/item?id=AST_1992__209__73_0> © Société mathématique de France, 1992, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CONGRUENCE SUBGROUP PROBLEM FOR ALGEBRAIC GROUPS: OLD AND NEW A. S. RAPINCHUK* Let G C GLn be an algebraic group defined over an algebraic number field K. Let 5 be a finite subset of the set VK of all valuations of K, containing the set V*£ of archimedean valuations. Denote by O(S) the ring of 5-integers in K and by GQ(S) the group of 5-units in G. To any nonzero ideal a C O(S) there corresponds the congruence subgroup Go(s)(*) = {96 G0(s) \ 9 = En (mod a)} , which is a normal subgroup of finite index in GQ(S)- The initial statement of the Congruence Subgroup Problem was : (1) Does any normal subgroup of finite index in GQ(S) contain a suitable congruence subgroup Go(s)(a) ? In fact, it was found by F. Klein as far back as 1880 that for the group SL2(Z) the answer to question (1) is "no". -
Chapter 2 Affine Algebraic Geometry
Chapter 2 Affine Algebraic Geometry 2.1 The Algebraic-Geometric Dictionary The correspondence between algebra and geometry is closest in affine algebraic geom- etry, where the basic objects are solutions to systems of polynomial equations. For many applications, it suffices to work over the real R, or the complex numbers C. Since important applications such as coding theory or symbolic computation require finite fields, Fq , or the rational numbers, Q, we shall develop algebraic geometry over an arbitrary field, F, and keep in mind the important cases of R and C. For algebraically closed fields, there is an exact and easily motivated correspondence be- tween algebraic and geometric concepts. When the field is not algebraically closed, this correspondence weakens considerably. When that occurs, we will use the case of algebraically closed fields as our guide and base our definitions on algebra. Similarly, the strongest and most elegant results in algebraic geometry hold only for algebraically closed fields. We will invoke the hypothesis that F is algebraically closed to obtain these results, and then discuss what holds for arbitrary fields, par- ticularly the real numbers. Since many important varieties have structures which are independent of the field of definition, we feel this approach is justified—and it keeps our presentation elementary and motivated. Lastly, for the most part it will suffice to let F be R or C; not only are these the most important cases, but they are also the sources of our geometric intuitions. n Let A denote affine n-space over F. This is the set of all n-tuples (t1,...,tn) of elements of F. -
Master's Thesis
MASTER'S THESIS Title of the Master's Thesis Engel Lie algebras submitted by Thimo Maria Kasper, BSc in partial fulfilment of the requirements for the degree of Master of Science (MSc) Vienna, 2018 Degree programme code: A 066 821 Degree programme: Master Mathematics Supervisor: Prof. Dr. Dietrich Burde Summary The purpose of this thesis is to present an investigation of Engel-n Lie algebras. In addition to the defining relations of Lie algebras these satisfy the so-called Engel-n identity ad(x)n = 0 for all x. Engel Lie algebras arise in the study of the Restricted Burnside Problem, which was solved by Efim Zelmanov in 1991. Beside a general introduction to the topic, special interest is taken in the exploration of the nilpotency classes of Engel-n Lie algebras for small values of n. At this, the primary objective is to elaborate the case of n = 3 explicitly. Chapter 1 concerns the general theory of Lie algebras. In the course of this, the essential properties of solvability and nilpotency are explained as they will be central in the subsequent discussion. Further, the definition of free Lie algebras which contributes to the establishment of the concept of free-nilpotent Lie algebras. In the last section several notions of group theory are surveyed. These will be useful in Chapter 2. The second chapter explains the origin and solution of the Burnside Problems. In that process, a historical survey on William Burnside and the first results on his fundamental questions are given. Next, the so-called Restricted Burnside Problem is considered and an overview of the most important steps to the solution is dis- played.