Congruences between the coefficients of the Tate curve via formal groups

Broumas, Antonios

doi:10.1090/S0002-9939-99-05133-3

Congruences between the coefficients of the Tate curve via formal groups
HTML articles powered by AMS MathViewer

by Antonios Broumas

Proc. Amer. Math. Soc. 128 (2000), 677-681

DOI: https://doi.org/10.1090/S0002-9939-99-05133-3

Published electronically: July 6, 1999

PDF | Request permission

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

A. Broumas, The invariant $A$ and the moduli problem Ig$(p^n)$, in preparation.
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) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 143–316 (French). MR 0337993
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
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
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, DOI 10.1515/9781400881710
Peter Roquette, Analytic theory of elliptic functions over local fields, Hamburger Mathematische Einzelschriften (N.F.), Heft 1, Vandenhoeck & Ruprecht, Göttingen, 1970. MR 0260753
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Dinesh S. Thakur, Automata-style proof of Voloch’s result on transcendence, J. Number Theory 58 (1996), no. 1, 60–63. MR 1387722, DOI 10.1006/jnth.1996.0061