Moduli Spaces and Arithmetic Dynamics
| Author | : Joseph H. Silverman |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 151 |
| Release | : |
| Genre | : Mathematics |
| ISBN | : 0821885030 |
Moduli Spaces And Arithmetic Dynamics
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Advanced Topics in the Arithmetic of Elliptic Curves
| Author | : Joseph H. Silverman |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 482 |
| Release | : 2013-12-01 |
| Genre | : Mathematics |
| ISBN | : 1461208513 |
In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this book: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.
Moduli Spaces of Riemann Surfaces
| Author | : Benson Farb |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 371 |
| Release | : 2013-08-16 |
| Genre | : Mathematics |
| ISBN | : 0821898876 |
Mapping class groups and moduli spaces of Riemann surfaces were the topics of the Graduate Summer School at the 2011 IAS/Park City Mathematics Institute. This book presents the nine different lecture series comprising the summer school, covering a selection of topics of current interest. The introductory courses treat mapping class groups and Teichmüller theory. The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of Torelli groups, and arithmetic mapping class groups. The courses consist of a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The book should be a valuable resource for graduate students and researchers interested in the topology, geometry and dynamics of moduli spaces of Riemann surfaces and related topics. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.
Moduli Spaces and Arithmetic Dynamics
| Author | : Joseph H. Silverman |
| Publisher | : Amer Mathematical Society |
| Total Pages | : 140 |
| Release | : 2012 |
| Genre | : Mathematics |
| ISBN | : 9780821875827 |
This monograph studies moduli problems associated to algebraic dynamical systems. It is an expanded version of the notes for a series of lectures delivered at a workshop on Moduli Spaces and the Arithmetic of Dynamical Systems at the Bellairs Research Institute, Barbados, in 2010. The author's goal is to provide an overview, with enough details and pointers to the existing literature, to give the reader an entry into this exciting area of current research. Topics covered include: (1) Construction and properties of dynamical moduli spaces for self-maps of projective space. (2) Dynatomic modular curves for the space of quadratic polynomials. (3) The theory of canonical heights associated to dynamical systems. (4) Special loci in dynamical moduli spaces, in particular the locus of post-critically finite maps. (5) Field of moduli and fields of definition for dynamical systems.
The Moduli Space of Curves
| Author | : Robert H. Dijkgraaf |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 570 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461242649 |
The moduli space Mg of curves of fixed genus g – that is, the algebraic variety that parametrizes all curves of genus g – is one of the most intriguing objects of study in algebraic geometry these days. Its appeal results not only from its beautiful mathematical structure but also from recent developments in theoretical physics, in particular in conformal field theory.
Dynamical Aspects of Teichmüller Theory
| Author | : Carlos Matheus Silva Santos |
| Publisher | : Springer |
| Total Pages | : 132 |
| Release | : 2018-07-09 |
| Genre | : Mathematics |
| ISBN | : 3319921592 |
This book is a remarkable contribution to the literature on dynamical systems and geometry. It consists of a selection of work in current research on Teichmüller dynamics, a field that has continued to develop rapidly in the past decades. After a comprehensive introduction, the author investigates the dynamics of the Teichmüller flow, presenting several self-contained chapters, each addressing a different aspect on the subject. The author includes innovative expositions, all the while solving open problems, constructing examples, and supplementing with illustrations. This book is a rare find in the field with its guidance and support for readers through the complex content of moduli spaces and Teichmüller Theory. The author is an internationally recognized expert in dynamical systems with a talent to explain topics that is rarely found in the field. He has created a text that would benefit specialists in, not only dynamical systems and geometry, but also Lie theory and number theory.
Birational Geometry and Moduli Spaces
| Author | : Elisabetta Colombo |
| Publisher | : Springer Nature |
| Total Pages | : 204 |
| Release | : 2020-02-25 |
| Genre | : Mathematics |
| ISBN | : 303037114X |
This volume collects contributions from speakers at the INdAM Workshop “Birational Geometry and Moduli Spaces”, which was held in Rome on 11–15 June 2018. The workshop was devoted to the interplay between birational geometry and moduli spaces and the contributions of the volume reflect the same idea, focusing on both these areas and their interaction. In particular, the book includes both surveys and original papers on irreducible holomorphic symplectic manifolds, Severi varieties, degenerations of Calabi-Yau varieties, uniruled threefolds, toric Fano threefolds, mirror symmetry, canonical bundle formula, the Lefschetz principle, birational transformations, and deformations of diagrams of algebras. The intention is to disseminate the knowledge of advanced results and key techniques used to solve open problems. The book is intended for all advanced graduate students and researchers interested in the new research frontiers of birational geometry and moduli spaces.
The Arithmetic of Dynamical Systems
| Author | : J.H. Silverman |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 518 |
| Release | : 2007-06-06 |
| Genre | : Mathematics |
| ISBN | : 0387699031 |
This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs.This graduate-level text provides an entry for students into an active field of research and serves as a standard reference for researchers.
The Geometry of Moduli Spaces of Sheaves
| Author | : Daniel Huybrechts |
| Publisher | : Cambridge University Press |
| Total Pages | : 345 |
| Release | : 2010-05-27 |
| Genre | : Mathematics |
| ISBN | : 1139485822 |
This edition has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces. The authors review changes in the field and point the reader towards further literature. An ideal text for graduate students or mathematicians with a background in algebraic geometry.
Arakelov Geometry and Diophantine Applications
| Author | : Emmanuel Peyre |
| Publisher | : Springer Nature |
| Total Pages | : 469 |
| Release | : 2021-03-10 |
| Genre | : Mathematics |
| ISBN | : 3030575594 |
Bridging the gap between novice and expert, the aim of this book is to present in a self-contained way a number of striking examples of current diophantine problems to which Arakelov geometry has been or may be applied. Arakelov geometry can be seen as a link between algebraic geometry and diophantine geometry. Based on lectures from a summer school for graduate students, this volume consists of 12 different chapters, each written by a different author. The first chapters provide some background and introduction to the subject. These are followed by a presentation of different applications to arithmetic geometry. The final part describes the recent application of Arakelov geometry to Shimura varieties and the proof of an averaged version of Colmez's conjecture. This book thus blends initiation to fundamental tools of Arakelov geometry with original material corresponding to current research. This book will be particularly useful for graduate students and researchers interested in the connections between algebraic geometry and number theory. The prerequisites are some knowledge of number theory and algebraic geometry.
