Arithmetic and GeometryTen Years in Alpbach (AMS-202)

Arithmetic and GeometryTen Years in Alpbach (AMS-202)

Gisbert Wüstholz and Clemens Fuchs

Print publication date: 2020

ISBN: 9780691193779

Publisher: Princeton University Press

Abstract

This book 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 heights 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 contains 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 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 originates in the Chowla–Selberg formula and relates values of the L-function for elliptic curves with the height of Heegner points on the curves. It proves the Gross–Zagier formula on Shimura curves and verifies the Colmez conjecture on average.

Table of Contents

Front Matter

Chapter One Introduction

Gisbert Wüstholz and Clemens Fuchs

Chapter Three Hyperelliptic Continued Fractions and Generalized Jacobians: Minicourse Given by Umberto Zannier

Laura Capuano, Peter Jossen, Christina Karolus, and Francesco Veneziano