The analytic order of III for modular elliptic curves

J. E. Cremona

Journal de théorie des nombres de Bordeaux (1993)

  • Volume: 5, Issue: 1, page 179-184
  • ISSN: 1246-7405

Abstract

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In this note we extend the computations described in [4] by computing the analytic order of the Tate-Shafarevich group III for all the curves in each isogeny class ; in [4] we considered the strong Weil curve only. While no new methods are involved here, the results have some interesting features suggesting ways in which strong Weil curves may be distinguished from other curves in their isogeny class.

How to cite

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Cremona, J. E.. "The analytic order of III for modular elliptic curves." Journal de théorie des nombres de Bordeaux 5.1 (1993): 179-184. <http://eudml.org/doc/93571>.

@article{Cremona1993,
abstract = {In this note we extend the computations described in [4] by computing the analytic order of the Tate-Shafarevich group III for all the curves in each isogeny class ; in [4] we considered the strong Weil curve only. While no new methods are involved here, the results have some interesting features suggesting ways in which strong Weil curves may be distinguished from other curves in their isogeny class.},
author = {Cremona, J. E.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {order of the Tate-Shafarevich group; isogeny},
language = {eng},
number = {1},
pages = {179-184},
publisher = {Université Bordeaux I},
title = {The analytic order of III for modular elliptic curves},
url = {http://eudml.org/doc/93571},
volume = {5},
year = {1993},
}

TY - JOUR
AU - Cremona, J. E.
TI - The analytic order of III for modular elliptic curves
JO - Journal de théorie des nombres de Bordeaux
PY - 1993
PB - Université Bordeaux I
VL - 5
IS - 1
SP - 179
EP - 184
AB - In this note we extend the computations described in [4] by computing the analytic order of the Tate-Shafarevich group III for all the curves in each isogeny class ; in [4] we considered the strong Weil curve only. While no new methods are involved here, the results have some interesting features suggesting ways in which strong Weil curves may be distinguished from other curves in their isogeny class.
LA - eng
KW - order of the Tate-Shafarevich group; isogeny
UR - http://eudml.org/doc/93571
ER -

References

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  1. [1] B. J. Birch and W. Kuyk (eds.), Modular Functions of One Variable IV, Lecture Notes in Mathematics, 476, Springer-Verlag (1975). Zbl0315.14014MR376533
  2. [2] A. Brumer and O. McGuinness, The behaviour of the Mordell-Weil group of elliptic curves, Bull. AMS (New Series)23 (1990), 375-382. Zbl0741.14010MR1044170
  3. [3] J.W.S. Cassels, Arithmetic on curves of genus 1 (VIII). On the conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math.217 (1965), 180-189. Zbl0241.14017MR179169
  4. [4] J.E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press1992. Zbl0758.14042MR1201151
  5. [5] V.I. Kolyvagin, Finiteness of E(Q) and IIIE/Q for a subclass of Weil curves, Math. USSR Izvest.32 (1989), 523-542. Zbl0662.14017MR954295
  6. [6] J. Vélu, Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris, sér. A273 (1971), 238-241. Zbl0225.14014MR294345

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