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Proceedings of the American Mathematical Society
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
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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.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05133-3
PII: S 0002-9939(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