Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Congruences between the coefficients
of the Tate curve via formal groups


Author: Antonios Broumas
Journal: Proc. Amer. Math. Soc. 128 (2000), 677-681
MSC (1991): Primary 11F33; Secondary 11G07, 14G20
Published electronically: July 6, 1999
MathSciNet review: 1641657
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $E_q:Y^2+XY = X^3 + h_4 X + h_6$ be the Tate curve with canonical differential, $\omega = dX/(2Y+X)$. If the characteristic is $p>0$, then the Hasse invariant, $H$, of the pair $(E_q,\omega)$ should equal one. If $p>3$, then calculation of $H$ leads to a nontrivial separable relation between the coefficients $h_4$ and $h_6$. If $p =2$ or $p =3$, Thakur related $h_4$ and $h_6$ via elementary methods and an identity of Ramanujan. Here, we treat uniformly all characteristics via explicit calculation of the formal group law of $E_q$. Our analysis was motivated by the study of the invariant $A$ which is an infinite Witt vector generalizing the Hasse invariant.


References [Enhancements On Off] (What's this?)

  • [B] A. Broumas, The invariant $A$ and the moduli problem Ig$(p^n)$, in preparation.
  • [DR] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 143–316. Lecture Notes in Math., Vol. 349 (French). MR 0337993
  • [D] Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197–272 (German). MR 0005125
  • [H] Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506881
  • [KM] Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569
  • [R] Peter Roquette, Analytic theory of elliptic functions over local fields, Hamburger Mathematische Einzelschriften (N.F.), Heft 1, Vandenhoeck & Ruprecht, Göttingen, 1970. MR 0260753
  • [S] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210
  • [T] Dinesh S. Thakur, Automata-style proof of Voloch’s result on transcendence, J. Number Theory 58 (1996), no. 1, 60–63. MR 1387722, 10.1006/jnth.1996.0061

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 11F33, 11G07, 14G20

Retrieve articles in all journals with MSC (1991): 11F33, 11G07, 14G20


Additional Information

Antonios Broumas
Affiliation: Mathematical Sciences Research Institute, 1000 Centennial Dr., Berkeley, California 94720
Email: antonios_m@yahoo.com

DOI: https://doi.org/10.1090/S0002-9939-99-05133-3
Keywords: Tate curve, Hasse invariant, formal group, $p$-typical, invariant $A$
Received by editor(s): April 27, 1998
Published electronically: July 6, 1999
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1999 American Mathematical Society