On the Hasse invariants of the Tate normal forms E5 and E7

Abstract

A formula is proved for the number of linear factors over Fl of the Hasse invariant of the Tate normal form E5(b) for a point of order 5, as a polynomial in the parameter b, in terms of the class number of the imaginary quadratic eld K = Q(p􀀀l), proving a conjecture of the author from 2005. A similar theorem is proved for quadratic factors with constant term 􀀀1, and a theorem is stated for the number of quartic factors of a speci c form in terms of the class number of Q(p 􀀀5l). These results are shown to imply a recent conjecture of Nakaya on the number of linear factors over Fl of the supersingular polynomial ss(5 ) l (X) corresponding to the Fricke group 􀀀 0 (5). The degrees and forms of the irreducible factors of the Hasse invariant of the Tate normal form E7 for a point of order 7 are determined, which is used to show that the polynomial ss(N ) l (X) for the group 􀀀 0 (N) has roots in Fl2 , for any prime l 6= N, when N 2 f2; 3; 5; 7g.