Book contents
- Frontmatter
- Contents
- Preface
- Heegner Points: The Beginnings
- Correspondence
- The Gauss Class Number Problem for Imaginary Quadratic Fields
- Heegner Points and Representation Theory
- Gross–Zagier Revisited
- Special Value Formulae for Rankin L-Functions
- Gross-Zagier Formula for GL(2), II
- Special Cycles and Derivatives of Eisenstein Series
- Faltings' Height and the Derivatives of Eisenstein Series
- Elliptic Curves and Analogies Between Number Fields and Function Fields
- Heegner Points and Elliptic Curves of Large Rank over Function Fields
- Periods and Points Attached to Quadratic Algebras
Elliptic Curves and Analogies Between Number Fields and Function Fields
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- Heegner Points: The Beginnings
- Correspondence
- The Gauss Class Number Problem for Imaginary Quadratic Fields
- Heegner Points and Representation Theory
- Gross–Zagier Revisited
- Special Value Formulae for Rankin L-Functions
- Gross-Zagier Formula for GL(2), II
- Special Cycles and Derivatives of Eisenstein Series
- Faltings' Height and the Derivatives of Eisenstein Series
- Elliptic Curves and Analogies Between Number Fields and Function Fields
- Heegner Points and Elliptic Curves of Large Rank over Function Fields
- Periods and Points Attached to Quadratic Algebras
Summary
Abstract. Well-known analogies between number fields and function fields have led to the transposition of many problems from one domain to the other. In this paper, we discuss traffic of this sort, in both directions, in the theory of elliptic curves. In the first part of the paper, we consider various works on Heegner points and Gross–Zagier formulas in the function field context; these works lead to a complete proof of the conjecture of Birch and Swinnerton-Dyer for elliptic curves of analytic rank at most 1 over function fields of characteristic > 3. In the second part of the paper, we review the fact that the rank conjecture for elliptic curves over function fields is now known to be true, and that the curves which prove this have asymptotically maximal rank for their conductors. The fact that these curves meet rank bounds suggests interesting problems on elliptic curves over number fields, cyclotomic fields, and function fields over number fields. These problems are discussed in the last four sections of the paper.
Introduction
The purpose of this paper is to discuss some work on elliptic curves over function fields inspired by the Gross–Zagier theorem and to present new ideas about ranks of elliptic curves from the function field case which I hope will inspire work over number fields.
We begin in Section 2 by reviewing the current state of knowledge on the conjecture of Birch and Swinnerton-Dyer for elliptic curves over function fields.
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- Heegner Points and Rankin L-Series , pp. 285 - 316Publisher: Cambridge University PressPrint publication year: 2004
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