Skip to main content
×
Home
    • Aa
    • Aa
  • Access
  • Cited by 3
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Balakrishnan, Jennifer S. 2016. On 3-adic heights on elliptic curves. Journal of Number Theory, Vol. 161, p. 119.


    Radziwiłł, Maksym and Soundararajan, K. 2015. Moments and distribution of central $$L$$ L -values of quadratic twists of elliptic curves. Inventiones mathematicae, Vol. 202, Issue. 3, p. 1029.


    Creutz, Brendan and Miller, Robert L. 2012. Second isogeny descents and the Birch and Swinnerton-Dyer conjectural formula. Journal of Algebra, Vol. 372, p. 673.


    ×
  • LMS Journal of Computation and Mathematics, Volume 14
  • November 2011, pp. 327-350

Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one

  • Robert L. Miller (a1) (a2)
  • DOI: http://dx.doi.org/10.1112/S1461157011000180
  • Published online: 01 November 2011
Abstract
Abstract

We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we rigorously prove the formula for 16714 of the 16725 such curves of conductor less than 5000.

    • Send article to Kindle

      To send this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one
      Your Kindle email address
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one
      Available formats
      ×
Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]A. Agashe , K. Ribet and W. A. Stein , ‘The Manin constant’, Pure Appl. Math. Q. 2 (2006) no. 2, 617636 part 2; MR 2251484(2007c:11076).

[2]A. Agashe and W. Stein , ‘Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero’, Math. Comp. 74 (2005) no. 249, 455484.

[6]C. Breuil , B. Conrad , F. Diamond and R. Taylor , ‘On the modularity of elliptic curves over ℚ: wild 3-adic exercises’, J. Amer. Math. Soc. 14 (2001) no. 4, 843939.

[7]J. P. Buhler , B. H. Gross and D. B. Zagier , ‘On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3’, Math. Comp. 44 (1985) no. 170, 473481; MR 777279(86g:11037).

[8]D. Bump , S. Friedberg and J. Hoffstein , ‘Non-vanishing theorems for L-functions of modular forms and their derivatives’, Invent. Math. 102 (1990) no. 3, 543618.

[12]B. Cha , ‘Vanishing of some cohomology goups and bounds for the Shafarevich–Tate groups of elliptic curves’, J. Number Theory 111 (2005) 154178.

[15]J. Cremona , ‘The elliptic curve database for conductors to 130000’, Algorithmic number theory, Lecture Notes in Computer Science 4076 (Springer, Berlin, 2006) 1129; MR 2282912(2007k:11087).

[16]J. E. Cremona and T. A. Fisher , ‘On the equivalence of binary quartics’, J. Symbolic Comput. 44 (2009) no. 6, 673682.

[17]J. E. Cremona and B. Mazur , ‘Visualizing elements in the Shafarevich–Tate group’, Experiment. Math. 9 (2000) no. 1, 1328.

[18]J. E. Cremona , M. Prickett and S. Siksek , ‘Height difference bounds for elliptic curves over number fields’, J. Number Theory 116 (2006) no. 1, 4268.

[21]B. Edixhoven , ‘On the Manin constants of modular elliptic curves’, Arithmetic algebraic geometry (Texel, 1989) (Birkhäuser, Boston, MA, 1991) 2539.

[23]T. Fisher , ‘Finding rational points on elliptic curves using 6-descent and 12-descent’, J. Algebra 320 (2008) no. 2, 853884.

[24]E. V. Flynn , F. Leprévost , E. F. Schaefer , W. A. Stein , M. Stoll and J. L. Wetherell , ‘Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves’, Math. Comp. 70 (2001) no. 236, 16751697; MR 1836926(2002d:11072)(electronic).

[25]R. Greenberg and V. Vatsal , ‘On the Iwasawa invariants of elliptic curves’, Invent. Math. 142 (2000) no. 1, 1763; MR 1784796(2001g:11169).

[27]G. Grigorov , A. Jorza , S. Patrikis , W. Stein and C. Tarniţǎ , ‘Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves’, Math. Comp. 78 (2009) 23972425.

[28]B. H. Gross , ‘Kolyvagin’s work on modular elliptic curves’, L-functions and arithmetic (Durham, 1989), London Mathematical Society Lecture Note Series 153 (Cambridge University Press, Cambridge, UK, 1991) 235256.

[29]B. Gross and D. Zagier , ‘Heegner points and derivatives of L-series’, Invent. Math. 84 (1986) no. 2, 225320.

[30]G. Hochschild and J.-P. Serre , ‘Cohomology of group extensions’, Trans. Amer. Math. Soc. 74 (1953) 110134.

[31]D. Jetchev , ‘Global divisibility of Heegner points and Tamagawa numbers’, Compos. Math. 144 (2008) no. 4, 811826.

[32]J. W. Jones , ‘Iwasawa L-functions for multiplicative abelian varieties’, Duke Math. J. 59 (1989) no. 2, 399420; MR 1016896(90m:11094).

[35]S. Lang , Number theory. III, vol. 60 (Springer, 1991).

[36]K. Matsuno , ‘Finite Λ-submodules of Selmer groups of abelian varieties over cyclotomic ℤp-extensions’, J. Number Theory 99 (2003) no. 2, 415443; MR 1969183(2004c:11098).

[37]B. Mazur , J. Tate and J. Teitelbaum , ‘On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer’, Invent. Math. 84 (1986) no. 1, 148; MR 830037(87e:11076).

[41]M. R. Murty and V. K. Murty , ‘Mean values of derivatives of modular L-series’, Ann. of Math. (2) 133 (1991) no. 3, 447475.

[42]M. J. Razar , ‘The non-vanishing of L(1) for certain elliptic curves with no first descents’, Amer. J. Math. 96 (1974) 104126; MR 0360596(50#13044a).

[43]M. J. Razar , ‘A relation between the two-component of the Tate–Šafarevič group and L(1) for certain elliptic curves’, Amer. J. Math. 96 (1974) 127144; MR 0360597(50#13044b).

[44]K. Rubin , ‘Congruences for special values of L-functions of elliptic curves with complex multiplication’, Invent. Math. 71 (1983) no. 2, 339364.

[45]K. Rubin , ‘The main conjectures of Iwasawa theory for imaginary quadratic fields’, Invent. Math. 103 (1991) no. 1, 2568.

[46]E. F. Schaefer and M. Stoll , ‘How to do a p-descent on an elliptic curve’, Trans. Amer. Math. Soc. 356 (2004) 12091231.

[50]J. H. Silverman , Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 151 (Springer, New York, 1994).

[58]A. Werner , ‘Local heights on abelian varieties with split multiplicative reduction’, Compositio Math. 107 (1997) no. 3, 289317; MR 1458753(98c:14039).

[62]S.-W. Zhang , ‘Gross–Zagier formula for GL(2) II’, Heegner points and Rankin L-series, Mathematical Sciences Research Institute Publications 49 (Cambridge University Press, Cambridge, 2004) 191214.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords: