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Arithmetic and GeometryTen Years in Alpbach (AMS-202)$
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Gisbert Wüstholz and Clemens Fuchs

Print publication date: 2019

Print ISBN-13: 9780691193779

Published to Princeton Scholarship Online: May 2020

DOI: 10.23943/princeton/9780691193779.001.0001

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(p.1) Chapter One Introduction
Arithmetic and Geometry
Gisbert Wüstholz, Clemens Fuchs
Princeton University Press

This introductory chapter provides an overview of the three topics discussed in this book: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings heights and L-functions. These topics were covered during the Alpbach Summerschool 2016, the celebration of the tenth session with outstanding speakers covering very different research areas in arithmetic and Diophantine geometry. The first course was given by Peter Scholze on local Shimura varieties and features recent results concerning the local Langlands conjecture. It considers the unpublished theorem which states that for each local Shimura datum, there exists a so-called local Shimura variety, which is a (pro-)rigid analytic space. The second course was given by Umberto Zannier and deals with a rather classical theme but from a modern point of view. His course is on hyperelliptic continued fractions and generalized Jacobians, using the classical Pell equation as the starting point. The third course was given by Shou-Wu Zhang and originates in the famous Chowla–Selberg formula, which was taken up by Gross and Zagier in 1984 to relate values of the L-function for elliptic curves with the height of Heegner points on the curves. Building on this work, X. Yuan, Shou-Wu Zhang, and Wei Zhang succeeded in proving the Gross–Zagier formula on Shimura curves and shortly later they verified the Colmez conjecture on average. In the course, Zhang presents new interesting aspects of the formula.

Keywords:   Shimura varieties, generalized Jacobians, hyperelliptic continued fractions, Faltings heights, L-functions, Langlands conjecture, Pell equation, Chowla–Selberg formula, Colmez conjecture

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