论文标题
TATE普通表格的HASSE不变性$ e_5 $和$ \ mathbb {q}的类号
The Hasse invariant of the Tate normal form $E_5$ and the class number of $\mathbb{Q}(\sqrt{-5l})$
论文作者
论文摘要
结果表明，表格$ g（x）= x^4+ax^3+（11a+2）x^2-ax-ax+1 $的不可约四分之一因素的数量划分了特征$ l $中的tate formal $ e_5 $ e_5 $的hasse不变性。 $ \ mathbb {q}（\ sqrt {-5l}）$，当$ l \ equiv 2,3 $ modulo $ 5 $。当$ l \ equiv 1，4 $ modulo $ 5 $时，类似的结果也适用于$ g（x）$的不可约二次因素。这意味着超过$ \ mathbb {f} _p $的线性因子数量的公式。
It is shown that the number of irreducible quartic factors of the form $g(x) = x^4+ax^3+(11a+2)x^2-ax+1$ which divide the Hasse invariant of the Tate normal form $E_5$ in characteristic $l$ is a simple linear function of the class number $h(-5l)$ of the field $\mathbb{Q}(\sqrt{-5l})$, when $l \equiv 2,3$ modulo $5$. A similar result holds for irreducible quadratic factors of $g(x)$, when $l \equiv 1, 4$ modulo $5$. This implies a formula for the number of linear factors over $\mathbb{F}_p$ of the supersingular polynomial $ss_p^{(5*)}(x)$ corresponding to the Fricke group $Γ_0^*(5)$.
