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Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin consists of invited expository articles and new developments extending from Yuri I. Manin s outstanding contributions to algebra, algebraic geometry, number theory, algorithmic complexity, noncommutative geometry and mathematical physics.

Among Manin s numerous achievements are the proof of the functional analogue of the Mordell conjecture, the theory of Gauss--Manin connection, the proof with V. Iskovskikh of the nonrationality of smooth quartic threefolds, the theory of $p$-adic automorphic functions, and the...

* Contains a selection of articles exploring geometric approaches to problems in algebra, algebraic geometry and number theory

* The collection gives a representative sample of problems and most recent results in algebraic and arithmetic geometry

* Text can serve as an intense introduction for graduate students and those wishing to pursue research in algebraic and arithmetic geometry

One of the great successes of twentieth century mathematics has been the remarkable qualitative understanding of rational and integral points on curves, gleaned in part through the theorems of Mordell, Weil, Siegel, and Faltings. It has become clear that the study of rational and integral points has deep connections to other branches of mathematics: complex algebraic geometry, Galois and etale cohomology, transcendence theory and diophantine approximation, harmonic analysis, automorphic forms, and analytic number theory.

This text, which focuses on higher dimensional varieties,...

This text offers a collection of survey and research papers by leading specialists in the field documenting the current understanding of higher dimensional varieties. Recently, it has become clear that ideas from many branches of mathematics can be successfully employed in the study of rational and integral points. This book will be very valuable for researchers from these various fields who have an interest in arithmetic applications, specialists in arithmetic geometry itself, and graduate students wishing to pursue research in this area.

This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the study of lines and conics. From the modern standpoint, arithmetic is the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that arithmetic...

Eisenstein series are an essential ingredient in the spectral theory of automorphic forms and an important tool in the theory of L-functions. They have also been exploited extensively by number theorists for many arithmetic purposes. Bringing together contributions from areas that are not usually interacting with each other, this volume introduces diverse users of Eisenstein series to a variety of important applications. With this juxtaposition of perspectives, the reader obtains deeper insights into the arithmetic of Eisenstein series.

The central theme of the exposition...