Abstract A formula is proved for the number of linear factors over F l of the Hasse invariant of the Tate normal form E 5 ( b ) for a point of order 5, as a polynomial in the parameter b, in terms of the class number of the imaginary quadratic field K = Q ( − 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 specific form in terms of the class number of Q ( − 5 l ) . These results are shown to imply a recent conjecture of Nakaya on the number of linear factors over F l of the supersingular polynomial s s l ( 5 ⁎ ) ( X ) corresponding to the Fricke group Γ 0 ⁎ ( 5 ) . The degrees and forms of the irreducible factors of the Hasse invariant of the Tate normal form E 7 for a point of order 7 are determined, which is used to show that the polynomial s s l ( N ⁎ ) ( X ) for the group Γ 0 ⁎ ( N ) has roots in F l 2 , for any prime l ≠ N , when N ∈ { 2 , 3 , 5 , 7 } .

中文翻译：

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关于 Tate 范式 E5 和 E7 的 Hasse 不变量

摘要 证明了阶数 5 的 Tate 范式 E 5 ( b ) 的 Hasse 不变量 F l 上线性因子个数的公式，作为参数 b 中的多项式，以类数表示虚二次场 K = Q ( − l ) ，证明了作者 2005 年的猜想。 证明了常数项为 -1 的二次因子的类似定理，并给出了特定形式的四次因子的个数的定理就 Q ( − 5 l ) 的类数而言。这些结果表明，Nakaya 最近对 Fricke 群 Γ 0 ⁎ (5) 对应的超奇异多项式 ssl ( 5 ⁎ ) ( X ) 在 F l 上的线性因子数进行了猜想。确定阶点 7 的 Tate 范式 E 7 的哈斯不变量的不可约因子的程度和形式，