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## Mordell-Weil Groups and Generating Series

*Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang*

### in The Gross-Zagier Formula on Shimura Curves

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691155913
- eISBN:
- 9781400845644
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691155913.003.0003
- Subject:
- Mathematics, Number Theory

This chapter deals with Mordell–Weil groups and generating series. It first provides an overview of the basics on Shimura curves and abelian varieties parametrized by Shimura curves before ... More

## Introduction and Statement of Main Results

*Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang*

### in The Gross-Zagier Formula on Shimura Curves

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691155913
- eISBN:
- 9781400845644
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691155913.003.0001
- Subject:
- Mathematics, Number Theory

This chapter states the main result of this book regarding Shimura curves and abelian varieties as well as the main idea of the proof of a complete Gross–Zagier formula on quaternionic Shimura curves ... More

## The Gross-Zagier Formula on Shimura Curves

*Xinyi Yuan, Shou-wu Zhang, and Wei Zhang*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691155913
- eISBN:
- 9781400845644
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691155913.001.0001
- Subject:
- Mathematics, Number Theory

This comprehensive account of the Gross–Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The ... More

## Trace of the Generating Series

*Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang*

### in The Gross-Zagier Formula on Shimura Curves

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691155913
- eISBN:
- 9781400845644
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691155913.003.0004
- Subject:
- Mathematics, Number Theory

This chapter proves the theorem that asserts the modularity of the generating series and the theorem dealing with abelian varieties parametrized by Shimura curves. Before presenting the proofs, 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 Heights of CM Points

*Xinyi Yuan, Shou-Wu Zhang, and Wei Zhang*

### in The Gross-Zagier Formula on Shimura Curves

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691155913
- eISBN:
- 9781400845644
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691155913.003.0008
- Subject:
- Mathematics, Number Theory

This chapter computes the local heights and compares them with the derivatives computed before. It checks the theorem place by place and takes into account all the assumptions on the Schwartz ... More

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