# Advanced Topics in the Arithmetic of Elliptic Curves

## Joseph H. Silverman

###
Springer-Verlag – Graduate Texts in Mathematics 151

ISBN: 978-0387943282
– 1st ed.
– © 1994
– 548 + xiii pages

Math. Subj. Class: 11G05, 14H52

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This book continues the treatment of the arithmetic theory of elliptic
curves begun in the first volume. The book begins with the theory of
elliptic and modular functions for the full modular group Γ(1),
including a discussion of Hecke operators and the *L*-series associated
to cusp forms. This is followed by a detailed study of elliptic curves
with complex multiplication, their associated Grössencharacters and
*L*-series, and applications to the construction of abelian
extensions of quadratic imaginary fields. Next comes a treatment of
elliptic curves over function fields and elliptic surfaces, including
specialization theorems for heights and sections. This material serves
as a prelude to the theory of minimal models and Néron models of
elliptic curves, with a discussion of special fibers, conductors, and
Ogg's formula. Next comes a brief description of *q*-models for
elliptic curves over **C** and **R**, followed by Tate's theory
of *q*-models for elliptic curves with non-integral
*j*-invariant over *p*-adic fields. The book concludes with
the construction of canonical local height functions on elliptic
curves, including explicit formulas for both archimedean and
non-archimedean fields.

**Contents**

Chapter I: | Elliptic and Modular Functions |

Chapter II: | Complex Multiplication |

Chapter III: | Elliptic Surfaces |

Chapter IV: | The Néron Model |

Chapter V: | Elliptic Curves over Complete Fields |

Chapter VI: | Local Height Functions |

Appendix A: | Some Useful Tables |

Errata List

No book is ever free from error or incapable of being improved. I
would be delighted to receive comments, good or bad, and corrections
from readers. You can send mail to me at
jhs@math.brown.edu

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