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

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