Skip to main content
×
×
Home

Ramanujan and the Modular j-Invariant

  • Bruce C. Berndt (a1) and Heng Huat Chan (a2)
Abstract

A new infinite product tn was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan’s assertions about tn by establishing new connections between themodular jinvariant and Ramanujan’s cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, tn generates the Hilbert class field of . This shows that tn is a new class invariant according to H. Weber’s definition of class invariants.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@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. 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.

      Ramanujan and the Modular j-Invariant
      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 <service> account. Find out more about sending content to Dropbox.

      Ramanujan and the Modular j-Invariant
      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 <service> account. Find out more about sending content to Google Drive.

      Ramanujan and the Modular j-Invariant
      Available formats
      ×
Copyright
References
Hide All
[1] Berndt, B. C., Ramanujan's Notebooks, Part III. Springer-Verlag, New York, 1991.
[2] Berndt, B. C., Ramanujan's Notebooks, Part V. Springer-Verlag, New York, 1998.
[3] Berndt, B. C., Bhargava, S. and Garvan, F. G., Ramanujan's theories of elliptic functions to alternative bases. Trans. Amer.Math. Soc. 347 (1995), 41634244.
[4] Berndt, B. C., Chan, H. H. and Zhang, L.-C., Ramanujan's class invariants, Kronecker's limit formula, and modular equations. Trans. Amer.Math. Soc. 349 (1997), 21252173.
[5] Berndt, B. C. and Rankin, R. A., Ramanujan: Letters and Commentary. American Mathematical Society, Providence, RI; London Mathematical Society, London, 1995.
[6] Berwick, W. E., Modular invariants expressible in terms of quadratic and cubic irrationalities. Proc. London Math. Soc. 28 (1927), 5369.
[7] Birch, B. J., Weber's class invariants. Mathematika 16 (1969), 283294.
[8] Borwein, J. M. and Borwein, P. B., Pi and the AGM. John Wiley, New York, 1987.
[9] Borwein, J. M. and Borwein, P. B., A cubic counterpart of Jacobi's identity and the AGM. Trans. Amer.Math. Soc. 323 (1991), 691701.
[10] Borwein, J. M., Borwein, P. B. and Garvan, F. G., Some cubic identities of Ramanujan. Trans. Amer.Math. Soc. 343 (1994), 3547.
[11] Chan, H. H., On Ramanujan's cubic transformation formula for 2f1(1/3, 2/3; 1;z). Math. Proc. Camb. Phil. Soc. 124 (1998), 193204.
[12] Chan, H. H. and Lang, M.-L., On Ramanujan's modular equations and Atkin-Lehner involutions. Israel J. Math. 103 (1998), 116.
[13] Chan, H. H. and Liaw, W.-C., On Russell type modular equations. Canad. J. Math., to appear.
[14] Chandrasekharan, K., Elliptic Functions. Springer-Verlag, Berlin, 1985.
[15] Cox, D. A., Primes of the form x2 + ny2 . Wiley, New York, 1989.
[16] Deuring, M., Die Klassenk¨orper der komplexen Multiplikation. Enz. Math. Wiss. Band I2, Heft 10 Teil II, Stuttgart, 1958.
[17] Greenhill, A. G., Complex multiplication moduli of elliptic functions. Proc. London Math. Soc. 19(1887–88), 301364.
[18] Greenhill, A. G., Table of complex multiplication moduli. Proc. London Math. Soc. 21(1889–90), 403422.
[19] Greenhill, A. G., The Applications of Elliptic Functions. Dover, New York, 1959.
[20] Newman, M., Construction and application of a class of modular functions II. Proc. London Math. Soc. 9 (1959), 373387.
[21] Ramanujan, S., Notebooks. 2 vols., Tata Institute of Fundamental Research, Bombay, 1957.
[22] Russell, R., On kλ − k′λ′ modular equations. Proc. London Math. Soc. 19 (1887), 90111.
[23] Russell, R., On modular equations. Proc. LondonMath. Soc. 21 (1890), 351395.
[24] S¨ohngen, H., Zur komplexen Multiplikation.Math. Ann. 111 (1935), 302328.
[25] Stark, H. M., Class numbers of complex quadratic fields. In: Modular functions of one variable I (ed. W. Kuijk), Lecture Notes in Math. 320(1973), Springer-Verlag, 154174.
[26] Weber, H., Lehrbuch der Algebra, dritter Band. Chelsea, New York, 1961.
Recommend this journal

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

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed