Quasi-quadratic elliptic curve point counting using rigid cohomology

Let E$E$ be a nonsupersingular elliptic curve over the finite field with pⁿ$p^n$ elements. We present a deterministic algorithm that computes the zeta function and hence the number of points of such a curve E$E$ in time quasi-quadratic in n$n$. An older algorithm having the same time complexity uses the canonical lift of E$E$, whereas our algorithm uses rigid cohomology combined with a deformation approach. An implementation in small odd characteristic turns out to give very good results.