Some Problems of Unlikely Intersections in Arithmetic and Geometry

Some Problems of Unlikely Intersections in Arithmetic and Geometry
Author :
Publisher : Princeton University Press
Total Pages : 176
Release :
ISBN-10 : 9780691153711
ISBN-13 : 069115371X
Rating : 4/5 (11 Downloads)

Synopsis Some Problems of Unlikely Intersections in Arithmetic and Geometry by : Umberto Zannier

This book considers the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. More generally, the book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension. The book is an expansion of the Hermann Weyl Lectures delivered by Umberto Zannier at the Institute for Advanced Study in Princeton in May 2010. The book consists of four chapters and seven brief appendixes, the last six by David Masser. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this (by Masser and the author) to a relative case of the Manin-Mumford issue. The fourth chapter focuses on the Andr -Oort conjecture (outlining work by Pila).

Some Problems of Unlikely Intersections in Arithmetic and Geometry

Some Problems of Unlikely Intersections in Arithmetic and Geometry
Author :
Publisher : Princeton University Press
Total Pages : 175
Release :
ISBN-10 : 9781400842711
ISBN-13 : 1400842719
Rating : 4/5 (11 Downloads)

Synopsis Some Problems of Unlikely Intersections in Arithmetic and Geometry by : Umberto Zannier

This book considers the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. More generally, the book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension. The book is an expansion of the Hermann Weyl Lectures delivered by Umberto Zannier at the Institute for Advanced Study in Princeton in May 2010. The book consists of four chapters and seven brief appendixes, the last six by David Masser. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this (by Masser and the author) to a relative case of the Manin-Mumford issue. The fourth chapter focuses on the André-Oort conjecture (outlining work by Pila).

Arithmetic and Geometry

Arithmetic and Geometry
Author :
Publisher : Princeton University Press
Total Pages : 187
Release :
ISBN-10 : 9780691197548
ISBN-13 : 0691197547
Rating : 4/5 (48 Downloads)

Synopsis Arithmetic and Geometry by : Gisbert Wüstholz

Arithmetic and Geometry presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures—which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria—provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings height and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach. The first course, taught by Peter Scholze, contains his recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces. The second course, taught by Umberto Zannier, addresses the famous Pell equation—not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians. The third course, taught by Shou-Wu Zhang, originates in the Chowla–Selberg formula, which was taken up by Gross and Zagier to relate values of the L-function for elliptic curves with the height of Heegner points on the curves. Zhang, X. Yuan, and Wei Zhang prove the Gross–Zagier formula on Shimura curves and verify the Colmez conjecture on average.

O-Minimality and Diophantine Geometry

O-Minimality and Diophantine Geometry
Author :
Publisher : Cambridge University Press
Total Pages : 235
Release :
ISBN-10 : 9781316301067
ISBN-13 : 1316301060
Rating : 4/5 (67 Downloads)

Synopsis O-Minimality and Diophantine Geometry by : G. O. Jones

This collection of articles, originating from a short course held at the University of Manchester, explores the ideas behind Pila's proof of the Andre–Oort conjecture for products of modular curves. The basic strategy has three main ingredients: the Pila–Wilkie theorem, bounds on Galois orbits, and functional transcendence results. All of these topics are covered in this volume, making it ideal for researchers wishing to keep up to date with the latest developments in the field. Original papers are combined with background articles in both the number theoretic and model theoretic aspects of the subject. These include Martin Orr's survey of abelian varieties, Christopher Daw's introduction to Shimura varieties, and Jacob Tsimerman's proof via o-minimality of Ax's theorem on the functional case of Schanuel's conjecture.

Algebraic Geometry: Salt Lake City 2015

Algebraic Geometry: Salt Lake City 2015
Author :
Publisher : American Mathematical Soc.
Total Pages : 658
Release :
ISBN-10 : 9781470435783
ISBN-13 : 1470435780
Rating : 4/5 (83 Downloads)

Synopsis Algebraic Geometry: Salt Lake City 2015 by : Richard Thomas

This is Part 2 of a two-volume set. Since Oscar Zariski organized a meeting in 1954, there has been a major algebraic geometry meeting every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle (2005). The American Mathematical Society has supported these summer institutes for over 50 years. Their proceedings volumes have been extremely influential, summarizing the state of algebraic geometry at the time and pointing to future developments. The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City, sponsored by the AMS with the collaboration of the Clay Mathematics Institute. This volume includes surveys growing out of plenary lectures and seminar talks during the meeting. Some present a broad overview of their topics, while others develop a distinctive perspective on an emerging topic. Topics span both complex algebraic geometry and arithmetic questions, specifically, analytic techniques, enumerative geometry, moduli theory, derived categories, birational geometry, tropical geometry, Diophantine questions, geometric representation theory, characteristic and -adic tools, etc. The resulting articles will be important references in these areas for years to come.

Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes)

Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes)
Author :
Publisher : World Scientific
Total Pages : 5393
Release :
ISBN-10 : 9789813272897
ISBN-13 : 9813272899
Rating : 4/5 (97 Downloads)

Synopsis Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes) by : Boyan Sirakov

The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.

Geometric Complex Analysis

Geometric Complex Analysis
Author :
Publisher : Springer
Total Pages : 359
Release :
ISBN-10 : 9789811316722
ISBN-13 : 9811316724
Rating : 4/5 (22 Downloads)

Synopsis Geometric Complex Analysis by : Jisoo Byun

The KSCV Symposium, the Korean Conference on Several Complex Variables, started in 1997 in an effort to promote the study of complex analysis and geometry. Since then, the conference met semi-regularly for about 10 years and then settled on being held biannually. The sixth and tenth conferences were held in 2002 and 2014 as satellite conferences to the Beijing International Congress of Mathematicians (ICM) and the Seoul ICM, respectively. The purpose of the KSCV Symposium is to organize the research talks of many leading scholars in the world, to provide an opportunity for communication, and to promote new researchers in this field.

The Arithmetic of Polynomial Dynamical Pairs

The Arithmetic of Polynomial Dynamical Pairs
Author :
Publisher : Princeton University Press
Total Pages : 252
Release :
ISBN-10 : 9780691235486
ISBN-13 : 0691235481
Rating : 4/5 (86 Downloads)

Synopsis The Arithmetic of Polynomial Dynamical Pairs by : Charles Favre

New mathematical research in arithmetic dynamics In The Arithmetic of Polynomial Dynamical Pairs, Charles Favre and Thomas Gauthier present new mathematical research in the field of arithmetic dynamics. Specifically, the authors study one-dimensional algebraic families of pairs given by a polynomial with a marked point. Combining tools from arithmetic geometry and holomorphic dynamics, they prove an “unlikely intersection” statement for such pairs, thereby demonstrating strong rigidity features for them. They further describe one-dimensional families in the moduli space of polynomials containing infinitely many postcritically finite parameters, proving the dynamical André-Oort conjecture for curves in this context, originally stated by Baker and DeMarco. This is a reader-friendly invitation to a new and exciting research area that brings together sophisticated tools from many branches of mathematics.

Point-Counting and the Zilber–Pink Conjecture

Point-Counting and the Zilber–Pink Conjecture
Author :
Publisher : Cambridge University Press
Total Pages : 267
Release :
ISBN-10 : 9781009170321
ISBN-13 : 1009170325
Rating : 4/5 (21 Downloads)

Synopsis Point-Counting and the Zilber–Pink Conjecture by : Jonathan Pila

Explores the recent spectacular applications of point-counting in o-minimal structures to functional transcendence and diophantine geometry.