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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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