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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


Modular curves, modular forms, lattices, Galois representations

Bas Edixhoven

in Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691142012
eISBN:
9781400839001
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142012.003.0002
Subject:
Mathematics, Number Theory

This chapter provides the necessary background concerning modular curves and modular forms. It covers modular curves, modular forms, lattices and modular forms, Galois representations attached to ... More


Approximating <i>V<sub>f</sub></i> over the complex numbers

Jean-Marc Couveignes

in Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691142012
eISBN:
9781400839001
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142012.003.0012
Subject:
Mathematics, Number Theory

This chapter addresses the problem of computing torsion divisors on modular curves with an application to the explicit calculation of modular representations. The final result of the chapter is ... More


Bounds for Arakelov invariants of modular curves

Bas Edixhoven and Robin de Jong

in Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691142012
eISBN:
9781400839001
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142012.003.0011
Subject:
Mathematics, Number Theory

This chapter gives bounds for all quantities on the right-hand side in the inequality in Theorems 9.1.1 and 9.2.5, in the context of the modular curves X₁(5l) with l > 5 prime, using the upper bounds ... More


Arithmetic Compactifications of PEL-Type Shimura Varieties

Kai-Wen Lan

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691156545
eISBN:
9781400846016
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691156545.001.0001
Subject:
Mathematics, Geometry / Topology

By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical ... 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


An upper bound for Green functions on Riemann surfaces

Franz Merkl

in Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691142012
eISBN:
9781400839001
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142012.003.0010
Subject:
Mathematics, Number Theory

This chapter deals with an upper bound for Green functions on Riemann surfaces.


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