Arithmetic and Geometry: Ten Years in Alpbach (AMS-202) (Annals of Mathematics Studies Book 371)

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.-- Source other than the Library of Congress

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 Cover Contents Preface 1. Introduction 2. Local Shimura Varieties: Minicourse Given by Peter Scholze 2.1 Introduction 2.2 Local Langlands Correspondence 2.3 Approach to LLC via Lubin-Tate Spaces 2.4 Approach to LLC via Rapoport-Zink Spaces 2.5 Some Basics of p-adic Geometry 2.6 Approach to LLC via Fargues-Fontaine Curve Bibliography 3. Hyperelliptic Continued Fractions and Generalized Jacobians: Minicourse Given by Umberto Zannier 3.1 Introduction and Some History 3.2 The Continued Fraction Expansion of Real Numbers 3.3 Continued Fractions in More General Settings 3.4 The Continued Fraction Expansion of Laurent Series3.5 Pell Equation in Polynomials 3.6 Distribution of Pellian Polynomials 3.7 The Pell Equation in the Nonsquarefree Case 3.8 A Skolem-Mahler-Lech Theorem for Algebraic Groups 3.9 Periodicity of the Degrees of the Partial Quotients 3.10 Solutions to the Exercises Bibliography 4. Faltings Heights and L-functions: Minicourse Given by Shou-Wu Zhang 4.1 Heights and L-functions 4.2 Shimura Curves and Averaged Colmez Conjecture 4.3 The Generalized Chowla-Selberg Formula 4.4 Higher Chowla-Selberg/Gross-Zagier Formula Bibliography "Lectures by outstanding scholars on progress made in the past ten years in the most progressive areas of arithmetic and geometry - primarily arithmetic geometry"-- Proporcionat per l'editor