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Arithmetic on Modular Curves

  • Glenn Stevens

Part of the Progress in Mathematics book series (PM, volume 20)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Glenn Stevens
    Pages 1-42
  3. Glenn Stevens
    Pages 43-75
  4. Glenn Stevens
    Pages 107-125
  5. Glenn Stevens
    Pages 126-165
  6. Glenn Stevens
    Pages 166-210
  7. Back Matter
    Pages 211-217

About this book

Introduction

One of the most intriguing problems of modern number theory is to relate the arithmetic of abelian varieties to the special values of associated L-functions. A very precise conjecture has been formulated for elliptic curves by Birc~ and Swinnerton-Dyer and generalized to abelian varieties by Tate. The numerical evidence is quite encouraging. A weakened form of the conjectures has been verified for CM elliptic curves by Coates and Wiles, and recently strengthened by K. Rubin. But a general proof of the conjectures seems still to be a long way off. A few years ago, B. Mazur [26] proved a weak analog of these c- jectures. Let N be prime, and be a weight two newform for r 0 (N) . For a primitive Dirichlet character X of conductor prime to N, let i\ f (X) denote the algebraic part of L (f , X, 1) (see below). Mazur showed in [ 26] that the residue class of Af (X) modulo the "Eisenstein" ideal gives information about the arithmetic of Xo (N). There are two aspects to his work: congruence formulae for the values Af(X) , and a descent argument. Mazur's congruence formulae were extended to r 1 (N), N prime, by S. Kamienny and the author [17], and in a paper which will appear shortly, Kamienny has generalized the descent argument to this case.

Keywords

algebra arithmetic function number theory proof

Authors and affiliations

  • Glenn Stevens
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4684-9165-4
  • Copyright Information Birkhäuser Boston 1982
  • Publisher Name Birkhäuser, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-8176-3088-1
  • Online ISBN 978-1-4684-9165-4
  • Series Print ISSN 0743-1643
  • Series Online ISSN 2296-505X
  • Buy this book on publisher's site
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