Arithmetic and Geometry : Ten Years in Alpbach (AMS-202) / Gisbert Wüstholz, Clemens Fuchs.

Format:

Book

Contributor:

Fuchs, Clemens, Editor.

Wüstholz, Gisbert, Editor.

Series:

Annals of mathematics studies ; number 202.

Princeton scholarship online.

Annals of Mathematics Studies ; 202

Language:

English

Subjects (All):

Arithmetical algebraic geometry.

Genre:

Conference papers and proceedings.

Physical Description:

1 online resource (187 pages).

Edition:

1st ed.

Place of Publication:

Princeton, NJ : Princeton University Press, [2019]

Language Note:

In English.

System Details:

Mode of access: World Wide Web.

Summary:

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.

Contents:

Frontmatter

Contents

Preface

1. Introduction

2. Local Shimura Varieties: Minicourse Given by Peter Scholze

3. Hyperelliptic Continued Fractions and Generalized Jacobians: Minicourse Given by Umberto Zannier

4. Faltings Heights and L-functions: Minicourse Given by Shou-Wu Zhang

List of Contributors

Notes:

Previously issued in print: 2019.

Includes bibliographical references.

Description based on online resource; title from PDF title page (publisher's Web site, viewed 06. Apr 2020)

ISBN:

9780691197548

0691197547

OCLC:

1125194243