Skip Navigation



The Quarterly Journal of Mathematics Advance Access published online on February 1, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hap001
This Article
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Shparlinski, I. E.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

TATE–SHAFAREVICH GROUPS AND FROBENIUS FIELDS OF REDUCTIONS OF ELLIPTIC CURVES

Igor E. Shparlinski{dagger}

Department of Computing, Macquarie University, Sydney, NSW 2109, Australia

{dagger} E-mail: igor{at}ics.mq.edu.au

Received 16 September 2008; Accepted for publication 5 January 2009.


  Abstract

Let E/Q be a fixed elliptic curve over Q which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, Cojocaru and Duke have obtained an asymptotic formula for the number of primes p≤x such that the reduction of E modulo p has a trivial Tate–Shafarevich group. Recent results of Cojocaru and David lead to a better error term. We introduce a new argument in the scheme of the proof, which gives a further improvement.


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.