A remark on Hecke operators and a theorem of Dwork and Koike

Abstract

Let $p\geq 5$ be prime, $\mathfrak{S}_p$ the set of all characteristic $p$ supersingular j-invariants in $\mathbb{F}_p-\{0,1728\}$, and $\mathfrak{M}_p$ the set of all monic irreducible quadratic polynomials in $\mathbb{F}_p[x]$ whose roots are supersingular j-invariants. A theorem of Dwork and Koike asserts that there are integers $A_p(\alpha),B_p(g),C_p(g)$, and a polynomial $D_p(x)\in\mathbb{F}_p[x]$ of degree $p-1$, for which

\begin{multline*} j(pz) \equiv j(z)^p + pD_p(j(z)) \\ + p\sum_{\alpha\in\mathfrak{S}_p}\frac{A_p(\alpha)}{j(z)-\alpha} + p \sum_{ g(x) \in \mathfrak{M}_p} \frac{B_p(g)j(z)+C_p(g)}{g(j(z))} \pmod{p^2}. \end{multline*}

It is natural to seek a description of the polynomials $D_p(x)$. Here we provide such a description in terms of certain Hecke polynomials.