Congruences between the coefficients of the Tate curve via formal groups

Let be the Tate curve with canonical differential, . If the characteristic is , then the Hasse invariant, , of the pair should equal one. If , then calculation of leads to a nontrivial separable relation between the coefficients and . If or , Thakur related and via elementary methods and an identity of Ramanujan. Here, we treat uniformly all characteristics via explicit calculation of the formal group law of . Our analysis was motivated by the study of the invariant which is an infinite Witt vector generalizing the Hasse invariant.