The Dynamical Mordell–Lang Conjecture
The Dynamical Mordell–Lang Conjecture

Author: Jason P. Bell

Publisher: American Mathematical Soc.

Published: 2016-04-20

Total Pages: 297

ISBN-13: 1470424088

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The Dynamical Mordell-Lang Conjecture is an analogue of the classical Mordell-Lang conjecture in the context of arithmetic dynamics. It predicts the behavior of the orbit of a point x under the action of an endomorphism f of a quasiprojective complex variety X. More precisely, it claims that for any point x in X and any subvariety V of X, the set of indices n such that the n-th iterate of x under f lies in V is a finite union of arithmetic progressions. In this book the authors present all known results about the Dynamical Mordell-Lang Conjecture, focusing mainly on a p-adic approach which provides a parametrization of the orbit of a point under an endomorphism of a variety.

The Dynamical Mordell-Lang Conjecture for Polynomial Endomorphisms of the Affine Plane
The Dynamical Mordell-Lang Conjecture for Polynomial Endomorphisms of the Affine Plane

Author: Junyi Xie

Publisher:

Published: 2017

Total Pages: 110

ISBN-13: 9782856298695

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In this paper we prove the Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane over the algebraic numbers. More precisely, let f be an endomorphism of the affine plan over the algebraic numbers. Let x be a point in the affine plan and C be a curve. If the intersection of C and the orbits of x is infinite, then C is periodic.

Number Theory – Diophantine Problems, Uniform Distribution and Applications
Number Theory – Diophantine Problems, Uniform Distribution and Applications

Author: Christian Elsholtz

Publisher: Springer

Published: 2017-05-26

Total Pages: 447

ISBN-13: 3319553577

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This volume is dedicated to Robert F. Tichy on the occasion of his 60th birthday. Presenting 22 research and survey papers written by leading experts in their respective fields, it focuses on areas that align with Tichy’s research interests and which he significantly shaped, including Diophantine problems, asymptotic counting, uniform distribution and discrepancy of sequences (in theory and application), dynamical systems, prime numbers, and actuarial mathematics. Offering valuable insights into recent developments in these areas, the book will be of interest to researchers and graduate students engaged in number theory and its applications.

Heights in Diophantine Geometry
Heights in Diophantine Geometry

Author: Enrico Bombieri

Publisher: Cambridge University Press

Published: 2006

Total Pages: 676

ISBN-13: 9780521712293

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This monograph is a bridge between the classical theory and modern approach via arithmetic geometry.

Rational Points, Rational Curves, and Entire Holomorphic Curves on Projective Varieties
Rational Points, Rational Curves, and Entire Holomorphic Curves on Projective Varieties

Author: Carlo Gasbarri

Publisher: American Mathematical Soc.

Published: 2015-12-22

Total Pages: 176

ISBN-13: 1470414589

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This volume contains papers from the Short Thematic Program on Rational Points, Rational Curves, and Entire Holomorphic Curves and Algebraic Varieties, held from June 3-28, 2013, at the Centre de Recherches Mathématiques, Université de Montréal, Québec, Canada. The program was dedicated to the study of subtle interconnections between geometric and arithmetic properties of higher-dimensional algebraic varieties. The main areas of the program were, among others, proving density of rational points in Zariski or analytic topology on special varieties, understanding global geometric properties of rationally connected varieties, as well as connections between geometry and algebraic dynamics exploring new geometric techniques in Diophantine approximation. This book is co-published with the Centre de Recherches Mathématiques.

The Mordell Conjecture
The Mordell Conjecture

Author: Hideaki Ikoma

Publisher: Cambridge University Press

Published: 2022-02-03

Total Pages: 179

ISBN-13: 1108845959

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This book provides a self-contained proof of the Mordell conjecture (Faltings's theorem) and a concise introduction to Diophantine geometry.

Mathematical Logic in the 20th Century
Mathematical Logic in the 20th Century

Author: Gerald E. Sacks

Publisher: World Scientific

Published: 2003

Total Pages: 712

ISBN-13: 9789812564894

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This invaluable book is a collection of 31 important both inideas and results papers published by mathematical logicians inthe 20th Century. The papers have been selected by Professor Gerald ESacks. Some of the authors are Gdel, Kleene, Tarski, A Robinson, Kreisel, Cohen, Morley, Shelah, Hrushovski and Woodin.

Advanced Topics in the Arithmetic of Elliptic Curves
Advanced Topics in the Arithmetic of Elliptic Curves

Author: Joseph H. Silverman

Publisher: Springer Science & Business Media

Published: 2013-12-01

Total Pages: 482

ISBN-13: 1461208513

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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.

Fundamentals of Diophantine Geometry
Fundamentals of Diophantine Geometry

Author: S. Lang

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 383

ISBN-13: 1475718101

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Diophantine problems represent some of the strongest aesthetic attractions to algebraic geometry. They consist in giving criteria for the existence of solutions of algebraic equations in rings and fields, and eventually for the number of such solutions. The fundamental ring of interest is the ring of ordinary integers Z, and the fundamental field of interest is the field Q of rational numbers. One discovers rapidly that to have all the technical freedom needed in handling general problems, one must consider rings and fields of finite type over the integers and rationals. Furthermore, one is led to consider also finite fields, p-adic fields (including the real and complex numbers) as representing a localization of the problems under consideration. We shall deal with global problems, all of which will be of a qualitative nature. On the one hand we have curves defined over say the rational numbers. Ifthe curve is affine one may ask for its points in Z, and thanks to Siegel, one can classify all curves which have infinitely many integral points. This problem is treated in Chapter VII. One may ask also for those which have infinitely many rational points, and for this, there is only Mordell's conjecture that if the genus is :;;; 2, then there is only a finite number of rational points.