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## Local Shimura Varieties: Minicourse Given by Peter Scholze

*Gisbert Wüstholz and Clemens Fuchs (eds)*

### in Arithmetic and Geometry: Ten Years in Alpbach

- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691193779
- eISBN:
- 9780691197548
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691193779.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses Peter Scholze's minicourse on local Shimura varieties. The goal of these lectures is to describe a program to construct local Langlands correspondence. The construction is ... More

## GEOMETRIC MODULAR FORMS

*Haruzo Hida*

### in Hilbert Modular Forms and Iwasawa Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198571025
- eISBN:
- 9780191718946
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571025.003.0004
- Subject:
- Mathematics, Algebra

This chapter discusses p-adic automorphic forms on Shimura variety from Springer, recalling the exact control theorem which connects the specialization (at a given weight) of the universal nearly ... More

## Shimura Varieties: A Hodge-Theoretic Perspective

*Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng (eds)*

### in Hodge Theory (MN-49)

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161341
- eISBN:
- 9781400851478
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161341.003.0012
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses certain cleverly constructed unions of modular varieties, called Shimura varieties, in the Hodge-theoretic perspective. The Shimura varieties can show the minimal (i.e., ... More

## About the André-Oort Conjecture

*Umberto Zannier*

### in Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153704
- eISBN:
- 9781400842711
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153704.003.0005
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses some aspects of the so-called André–Oort conjecture. It observes how this resembles the pattern of the statements discussed so far, recalling also a brief history. After a few ... More

## Arithmetic and Geometry: Ten Years in Alpbach (AMS-202)

*Gisbert Wüstholz and Clemens Fuchs (eds)*

- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691193779
- eISBN:
- 9780691197548
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691193779.001.0001
- Subject:
- Mathematics, Geometry / Topology

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 ... More

## Introduction

*Gisbert Wüstholz and Clemens Fuchs (eds)*

### in Arithmetic and Geometry: Ten Years in Alpbach (AMS-202)

- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691193779
- eISBN:
- 9780691197548
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691193779.003.0001
- Subject:
- Mathematics, Geometry / Topology

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 ... More

## Hodge Theory (MN-49)

*Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dung Tráng*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691161341
- eISBN:
- 9781400851478
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161341.001.0001
- Subject:
- Mathematics, Geometry / Topology

This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The ... More

## Integral models of local Shimura varieties

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0025
- Subject:
- Mathematics, Geometry / Topology

This chapter explains an application of the theory developed in these lectures towards the problem of understanding integral models of local Shimura varieties. As a specific example, it resolves ... More

## Local Shimura varieties

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0024
- Subject:
- Mathematics, Geometry / Topology

This chapter specializes the theory back to the case of local Shimura varieties, and explains the relation with Rapoport-Zink spaces. It begins with a local Shimura datum. A local Shimura datum is a ... More

## Introduction

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0001
- Subject:
- Mathematics, Geometry / Topology

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case ... More

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