Towards computing the canonical lift of an ordinary elliptic curve in medium characteristic

Satoh's algorithm for counting the number of points of an elliptic curve $E/\mathbb F_q$ with $q=p^n$ is the fastest known algorithm when $p$ is fixed: it computes the invertible eigenvalue $»$ of the Frobenius to $p$-adic precision $m$ in time $\tilde{O}(p^2 n m)$. Since by Hasse's bound, recovering $\chi_{\pi}$ requires working at precision $m=O(n)$, the point counting complexity is of $\tilde{O}(p^2 n^2)$, quasi-quadratic in the degree $n$. Unfortunately, the term $p^2$ in the complexity makes Satoh's algorithm suitable only for smaller $p$. For medium sized $p$, one can use Kedlaya's algorithm which cost $\tilde{O}(p n^2 m)$ or a variant by Harvey's which cost $\tilde{O}(p^{1/2} n^{5/2} m + n^4 m)$, which have a better complexity on $p$ but a worse one on $n$. For large $p$, the SEA algorithm costs $\tilde{O}(log^4 q)$. In this talk, we improve the dependency on $p$ of Satoh's algorithm while retaining the dependency on $n$ to bridge the gap towards medium characteristic. We develop a new algorithm with a complexity of $\tilde{O}(p n m)$. In the particular case where we are furthermore provided with a rational point of $p$-torsion, we even improve this complexity to $\tilde{O}(p^{1/2} n m)$. This is a joint work with Abdoulaye Maiga.