Elliptic Pseudoprimes

Abstract

Let E be an elliptic curve over Q with complex multiplication by an order in an imaginary quadratic field. Let ${\psi _n}$ denote the nth division polynomial, and let P be a rational point of E of infinite order. A natural number n is called an elliptic pseudoprime if $n|{\psi _{n + 1}}(P)$ and n is composite. Let $N(x)$ denote the number of elliptic pseudoprimes up to x. We show that $N(x) \ll x{(\log \log x)^{7/2}}/{(\log x)^{3/2}}$. More generally, if ${P_1}, \ldots ,{P_r}$ are r independent rational points of E which have infinite order, and $\Gamma$ is the subgroup generated by them, denote by ${N_\Gamma }(x)$ the number of composite $n \leq x$ satisfying $n|{\psi _{n + 1}}({P_i})$, $1 \leq i \leq r$. For $r \geq 2$, we prove ${N_\Gamma }(x) \ll x\exp ( - c\sqrt {(\log x)(\log \log x))}$ for some positive constant c.