On supersingular elliptic curves and hypergeometric functions (Q444616)

The paper deals with two families of elliptic curves, namely NEWLINE\[NEWLINE E_{\frac{1}{3}}(\lambda) : y^2+\lambda xy+\lambda^2y = x^3 \quad \lambda\neq 0,27 NEWLINE\]NEWLINE NEWLINE\[NEWLINE E_{\frac{1}{12}}(\lambda) : y^2 = 4x^3 -27\lambda x-27\lambda \quad \lambda\neq 0,1 NEWLINE\]NEWLINE (another family \(E_{\frac{1}{4}}(\lambda) : y^2=(x-1)(x^2+\lambda)\) is treated in a forthcoming paper of \textit{A. El-Guindy} and \textit{K. Ono}), and their supersingular locus NEWLINE\[NEWLINE S_{p,i}(\lambda)=\prod_{ {\lambda_0\in \overline{\mathbb{F}}_p} \atop {E_i(\lambda_0)\text{ is supersingular }} } \, (\lambda-\lambda_0) \qquad (i=\frac{1}{3},\frac{1}{12})\;. NEWLINE\]NEWLINE The author proves first that there are exactly \(\lfloor pi \rfloor\) distinct values of \(\lambda\) for which \(E_i(\lambda)\) is supersingular over \(\overline{\mathbb{F}}_p\,\). Then a careful computation of the Hasse invariants and some combinatorial identities provide a description of \(S_{p,i}(\lambda)\) in terms of the hypergeometric function NEWLINE\[NEWLINE \;_2F_1\left(\begin{matrix} a & b \\ \;& c \end{matrix}\Big| z \right) := \sum_{n\geq 0} \frac{(a)_n(b)_n}{(c)_n} \frac{z^n}{n!} NEWLINE\]NEWLINE truncated at \(p\) (where \((x)_n=x(x+1)\cdots(x+n-1)\,\)), so that the supersingular curves can be identified by the zeroes modulo \(p\) of such (truncated) functions. In particular, for \(p\geq 5\), the author proves that NEWLINE\[NEWLINE \begin{matrix} S_{p,\frac{1}{3}}(\lambda) & \equiv \lambda^{\lfloor\frac{p}{3}\rfloor} \;_2F_1\left(\begin{matrix} \frac{1}{3} & \frac{2}{3} \\ \;& 1 \end{matrix} \Big| \frac{27}{\lambda}\right)_p \pmod p \\ \;& \equiv \sum_{n=0}^{p-1} \frac{(\frac{1}{3})_n(\frac{2}{3})_n}{(1)_n} \left(\frac{27}{\lambda}\right)^n\frac{1}{n!} \pmod p \end{matrix} NEWLINE\]NEWLINE (the relations for \(S_{p,\frac{1}{12}}(\lambda)\) are similar but also depend on the congruence class of \(p\) modulo 12), a results which reminds of the well known one NEWLINE\[NEWLINE S_{p,\frac{1}{2}}(\lambda) \equiv \sum_{n=0}^{p-1} \frac{(\frac{1}{2})_n(\frac{1}{2})_n}{(1)_n} \frac{\lambda^n}{n!} \pmod p NEWLINE\]NEWLINE for the Legendre curves NEWLINE\[NEWLINE E_{\frac{1}{2}}(\lambda) : y^2=x(x-1)(x-\lambda) \quad \lambda\neq 0,1 \;.NEWLINE\]