IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-1-4612-1974-3_21.html
   My bibliography  Save this book chapter

Wiles’ Theorem and the Arithmetic of Elliptic Curves

In: Modular Forms and Fermat’s Last Theorem

Author

Listed:
  • Henri Darmon

Abstract

Thanks to the work of Wiles [Wi], completed by Taylor-Wiles [TW] and extended by Diamond [Di], we now know that all elliptic curves over the rationals (having good or semi-stable reduction at 3 and 5) are modular. This breakthrough has far-reaching consequences for the arithmetic of elliptic curves. As Mazur wrote in [Ma3], “It has been abundantly clear for years that one has a much more tenacious hold on the arithmetic of an elliptic curve E/Q if one supposes that it is […] parametrized [by a modular curve].” This expository article explores some of the implications of Wiles’ theorem for the theory of elliptic curves, with particular emphasis on the Birch and Swinnerton-Dyer conjecture, now the main outstanding problem in the field.

Suggested Citation

  • Henri Darmon, 1997. "Wiles’ Theorem and the Arithmetic of Elliptic Curves," Springer Books, in: Gary Cornell & Joseph H. Silverman & Glenn Stevens (ed.), Modular Forms and Fermat’s Last Theorem, pages 549-569, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-1974-3_21
    DOI: 10.1007/978-1-4612-1974-3_21
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a
    for a similarly titled item that would be available.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sprchp:978-1-4612-1974-3_21. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.