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Arithmetic on Curves
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 14, Number 2, April 1986 ARITHMETIC ON CURVES BY BARRY MAZUR CONTENTS I. What are Diophantine problems? 1. Miscellaneous Diophantine problems 2. Systematic formulations 3. Digression: Questions of integral solutions vs. Questions of Diophantine approxima tions II. Curves 1. Plane curves 2. Algebraic curves 3. Smooth curves 4. Riemann surfaces 5. Simplifying the singularities of a plane curve 6. Genus 7. The fundamental trichotomy 8. Interlude: A geometric finiteness result 9. On Mordell's conjecture and its analogue III. Jacobians and abelian varieties 1. An extension of the chord-and-tangent method to curves of genus > 2: The jacobian of a curve 2. The analytic parametrization of J by algebraic integrals modulo periods 3. The algebraic nature of the jacobian 4. The arithmetic nature of the jacobian IV. Classification of families with bounded bad reduction 1. Kodaira's problem 2. Shafarevich's problems 3. Parshin's construction: Shafarevich's conjecture implies Mordell's conjecture V. Heights 1. Height of points 2. Normed vector spaces 3. Heights of abelian varieties 4. A block diagram Received by the editors June 12, 1985. 1980Mathematics Subject Classification (1985 Revision). Primary 14Hxx, 14Kxx, llDxx, 00-01, 01-01, 01A65, 11-01, 11G05, 11G10, 11G15. ©1986 American Mathematical Society 0273-0979/86 $1.00 + $.25 per page 207 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 208 BARRY MAZUR Bibliography 1. General references 2. References requiring some background 3. Accounts of the proof of Mordell's conjecture which require familiarity with the techniques of number theory or algebraic geometry 4. -
Book-49693.Pdf
APPLICATIONS OF GROUP THEORY TO COMBINATORICS 2 SELECTED PAPERS FROM THE COM MAC CONFERENCE ON APPLICATIONS OF GROUP THEORY TO COMBINATORICS, POHANG, KOREA, 9–12 JULY 2007 Applications of Group Theory to Combinatorics Editors Jack Koolen Department of Mathematics, Pohang University of Science and Technology, Pohang 790–784, Korea Jin Ho Kwak Department of Mathematics, Pohang University of Science and Technology, Pohang 790–784, Korea Ming-Yao Xu Department of Mathematics, Peking University, Beijing 100891, P.R.China Supported by Com2MaC-KOSEF, Korea CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business © 2008 Taylor & Francis Group, London, UK Typeset by Vikatan Publishing Solutions (P) Ltd., Chennai, India Printed and bound in Great Britain by Anthony Rowe (A CPI-group Company), Chippenham, Wiltshire All rights reserved. No part of this publication or the information contained herein may be repro- duced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: CRC Press/Balkema P.O. Box 447, 2300 AK Leiden, The Netherlands e-mail: [email protected] www.crcpress.com – www.taylorandfrancis.co.uk – www.balkema.nl ISBN: 978-0-415-47184-8 (Hardback) ISBN: 978-0-203-88576-5 (ebook) Applications of Group Theory to Combinatorics - Koolen, Kwak & Xu (eds) © 2008 Taylor & Francis Group, London, ISBN 978-0-415-47184-8 Table of Contents Foreword VII About the editors IX Combinatorial and computational group-theoretic methods in the study of graphs, maps and polytopes with maximal symmetry 1 M. -
Complex Algebraic Varieties and Their Cohomology
Complex Algebraic Varieties and their Cohomology Donu Arapura July 16, 2003 Introduction In these notes, which originated from various “second courses” in algebraic geometry given at Purdue, I study complex algebraic varieties using a mixture of algebraic, analytic and topological methods. I assumed an understanding of basic algebraic geometry (around the level of [Hs]), but little else beyond stan- dard graduate courses in algebra, analysis and elementary topology. I haven’t attempted to prove everything, often I have been content to give a sketch along with a reference to the relevent section of Hartshorne [H] or Griffiths and Harris [GH]. These weighty tomes are, at least for algebraic geometers who came of age when I did, the canonical texts. They are a bit daunting however, and I hope these notes makes some of this material more accessible. These notes are pretty rough and somewhat incomplete at the moment. Hopefully, that will change with time. For updates check http://www.math.purdue.edu/∼dvb 1 Contents 1 Manifolds and Varieties via Sheaves 5 1.1 Sheaves of functions ......................... 5 1.2 Manifolds ............................... 6 1.3 Algebraic varieties .......................... 9 1.4 Stalks and tangent spaces ...................... 11 1.5 Nonsingular Varieties ......................... 13 1.6 Vector fields .............................. 14 2 Generalities about Sheaves 16 2.1 The Category of Sheaves ...................... 16 2.2 Exact Sequences ........................... 17 2.3 The notion of a scheme ....................... 19 2.4 Toric Varieties ............................ 20 2.5 Sheaves of Modules .......................... 22 2.6 Direct and Inverse images ...................... 24 3 Sheaf Cohomology 26 3.1 Flabby Sheaves ............................ 26 3.2 Cohomology .............................. 27 3.3 Soft sheaves ............................. -
Lectures on Modular Forms 1St Edition Free Download
FREE LECTURES ON MODULAR FORMS 1ST EDITION PDF Joseph J Lehner | 9780486821405 | | | | | Lectures on Modular Forms - Robert C. Gunning - Google книги In mathematicsa modular form is a complex analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular groupand also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topologysphere packingand string theory. Instead, modular functions are meromorphic that is, they are almost holomorphic except for a set of isolated points. Modular form theory is a special case of the more general theory of automorphic formsand therefore can now be seen as just the most concrete part of a rich theory of discrete groups. Modular forms can also be interpreted as sections of a specific line bundles on modular varieties. The dimensions of these spaces of modular forms can be computed using the Riemann—Roch theorem [2]. A Lectures on Modular Forms 1st edition form of weight k for the modular group. A modular form can equivalently be defined as a function F from the set of lattices in C to the set of complex numbers which satisfies certain conditions:. The simplest examples from this point of view are the Eisenstein series. Then E k is a modular form of weight k. An even unimodular lattice L in R n is a lattice generated by n Lectures on Modular Forms 1st edition forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in L is an even integer. -
Basic Theory of Abelian Varieties 1. Definitions
ABELIAN VARIETIES J.S. MILNE Abstract. These are the notes for Math 731, taught at the University of Michigan in Fall 1991, somewhat revised from those handed out during the course. They are available at www.math.lsa.umich.edu/∼jmilne/. Please send comments and corrections to me at [email protected]. v1.1 (July 27, 1998). First version on the web. These notes are in more primitive form than my other notes — the reader should view them as an unfinished painting in which some areas are only sketched in. Contents Introduction 1 Part I: Basic Theory of Abelian Varieties 1. Definitions; Basic Properties. 7 2. Abelian Varieties over the Complex Numbers. 9 3. Rational Maps Into Abelian Varieties 15 4. The Theorem of the Cube. 19 5. Abelian Varieties are Projective 25 6. Isogenies 29 7. The Dual Abelian Variety. 32 8. The Dual Exact Sequence. 38 9. Endomorphisms 40 10. Polarizations and Invertible Sheaves 50 11. The Etale Cohomology of an Abelian Variety 51 12. Weil Pairings 52 13. The Rosati Involution 53 14. The Zeta Function of an Abelian Variety 54 15. Families of Abelian Varieties 57 16. Abelian Varieties over Finite Fields 60 17. Jacobian Varieties 64 18. Abel and Jacobi 67 Part II: Finiteness Theorems 19. Introduction 70 20. N´eron models; Semistable Reduction 76 21. The Tate Conjecture; Semisimplicity. 77 c 1998 J.S. Milne. You may make one copy of these notes for your own personal use. 1 0J.S.MILNE 22. Geometric Finiteness Theorems 82 23. Finiteness I implies Finiteness II. 87 24. -
Bibliography
Bibliography 1. Abikoff, William, TherealanalytictheoryofTeichm¨uller space, Lecture Notes in Mathematics, 820, Springer-Verlag, Berlin, 1980. 2. Abramovich, D., Corti, A. and Vistoli, A., Twisted bundles and admissible covers, Comm. Algebra 31 (2003), 3547–3618. 3. Adem, A., Leida, J. and Ruan, Y., Orbifolds and stringy topology, Cambridge Tracts in Mathematics, 171, Cambridge University Press, New York, 2007. 4. Ahlfors, Lars V., Lectures on quasiconformal mappings,University Lecture Series, 38, American Mathematical Society, Providence, RI, 2006. 5. Ahlfors, Lars and Bers, Lipman, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. 6. Alexeev, Valery, Compactified Jacobians and Torelli map, Publ. Res. Inst. Math. Sci. 40 (2004), 1241–1265. 7. Altman, Allen B. and Kleiman, Steven L., Compactifying the Jacobian, Bull. Amer. Math. Soc. 82 (1976), 947–949. 8. Altman, Allen B. and Kleiman, Steven L., Compactifying the Picard scheme, Adv. in Math. 35 (1980), 50–112. 9. Altman, Allen B. and Kleiman, Steven L., Compactifying the Picard scheme. II, Amer. J. Math. 101 (1979), 10–41. 10. Andreotti, Aldo, On a theorem of Torelli, Amer. J. Math. 80 (1958), 801–828. 11. Aprodu, Marian, Brill–Noether theory for curves on K3 surfaces,in Contemporary geometry and topology and related topics, Cluj Univ. Press, Cluj-Napoca, 2008, pp. 1–12. 12. Aprodu, Marian and Farkas, Gavril, Koszul cohomology and applications to moduli, arXiv:0811.3117v1 [math.AG]. 13. Aprodu, Marian and Farkas, Gavril, Green’s conjecture for curves on arbitrary K3 surfaces, arXiv:0911.5310 (2009). 14. Aprodu, Marian and Nagel, Jan, Koszul cohomology and Algebraic Geometry, University Lecture Series, 52, American Mathematical Society, Providence, RI, 2010. -
Arxiv:1801.02479V1 [Math.AG] 8 Jan 2018 O Ealdmngahc Ntetpco Otemr Eets Recent More the to Or Topic [Kob98 the DR11]
HYPERBOLICITY NOTIONS FOR VARIETIES DEFINED OVER A NON-ARCHIMEDEAN FIELD RITA RODRÍGUEZ VÁZQUEZ Abstract. Firstly, we pursue the work of W. Cherry on the analogue of the Kobayashi semi distance dCK that he introduced for analytic spaces defined over a non-Archimedean metrized field k. We prove various char- acterizations of smooth projective varieties for which dCK is an actual distance. Secondly, we explore several notions of hyperbolicity for a smooth algebraic curve X defined over k. We prove a non-Archimedean analogue of the equivalence between having negative Euler characteristic and the normality of certain families of analytic maps taking values in X. Contents 1. Introduction 1 2. Berkovich analytic spaces 6 3. Smooth analytic curves 9 4. Cherry hyperbolicity 16 5. Zalcman’s reparametrization lemma 18 6. Further notions of hyperbolicity 22 7. Curves with non-negative Euler characteristic 28 8. Analytic maps on special domains 29 9. Proof of Theorem D 33 10. Proof of Theorem C 35 References 37 arXiv:1801.02479v1 [math.AG] 8 Jan 2018 1. Introduction The notion of Kobayashi hyperbolicity [Kob67] is arguably one of the fundamental notions in complex geometry. We refer to [Kob98] and [Lan87] for detailed monographics on the topic or to the more recent surveys [Voi03, DR11]. It is a fundamental and remarkable fact that hyperbolic compact complex spaces can be characterized as follows: Date: December 26, 2017. Research supported by the ERC grant Nonarcomp no. 307856. 1 2 RITA RODRÍGUEZ VÁZQUEZ Theorem 1. Let X be a smooth compact complex analytic space endowed with a hermitian metric. -
DIOPHANTINE GEOMETRY on CURVES OVER FUNCTION FIELDS These Are Extended Notes of My Five Hours Lecture at the Winter School on Al
DIOPHANTINE GEOMETRY ON CURVES OVER FUNCTION FIELDS CARLO GASBARRI These are extended notes of my five hours lecture at the Winter School on Algebraic Curves, Riemann Surfaces and Moduli Spaces held at the Morningside Center of Mathematics, CAS from march 4th to 8th 2019. We explain the main steps of the proof of three cornerstone theorems in the diophantine geometry of smooth projective curves over a function field in one variable in characteristic zero: Let F be the function field of a smooth projective curve B defined over C. { We prove that every XF curve of genus zero defined over F has infinitely many F {rational points and therefore it is isomorphic, over F , to P1. { We introduce then the important concept of isotrivality for curves defined over F : Isotriv- ial curves are those which, after a finite extension of F , are isomorphic to curves defined over C (which is naturally a subfield of F ). In particular we will prove an important isotriviality criterion as a consequence of a classical Theorem by De Franchis (which is proved in the notes). { We prove that a smooth projective curve XF of genus one, equipped with a rational point, has a natural structure of a commutative group variety. If the curve is not isotrivial, then we will prove that this group is finitely generated. This is known as the Mordell Weil Theorem. { We prove that a non isotrivial smooth projective curve XF of genus at least two has only finitely many F {rational points. We tried to keep the notes as much as self contained as possible. -
Lectures on Modular Forms 1St Edition Ebook
LECTURES ON MODULAR FORMS 1ST EDITION PDF, EPUB, EBOOK Joseph J Lehner | 9780486821405 | | | | | Lectures on Modular Forms 1st edition PDF Book Main article: Atkin—Lehner theory. A First Course in Modular Forms is written for beginning graduate students and advanced undergraduates. Siegel modular forms are associated to larger symplectic groups in the same way in which classical modular forms are associated to SL 2, R ; in other words, they are related to abelian varieties in the same sense that classical modular forms which are sometimes called elliptic modular forms to emphasize the point are related to elliptic curves. Five points determine a conic Projective line Rational normal curve Riemann sphere Twisted cubic. We call this lattice L n. Then E k is a modular form of weight k. Categories : Modular forms Analytic number theory Special functions. From Wikipedia, the free encyclopedia. Please help improve this section by adding citations to reliable sources. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. What is more, it can be endowed with the structure of a Riemann surface , which allows one to speak of holo- and meromorphic functions. Topics in algebraic curves. Main article: Cusp form. If we allow denominators rational functions instead of polynomials , we can let F be the ratio of two homogeneous polynomials of the same degree. Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell—Weil theorem Nagell—Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof —Elkies—Atkin algorithm.