In a cryptography book I read that people does not known how to compute the number of points on a Jacobian of a hyperelliptic curve $C$ over a finite field $F\_q$? Is this true? It seems easy to compute it knowing the eigenvalues of the Frobenius action on $H^1(C)$, which could be recovered knowing $\sharp C(F\_{q^l})$ for all $l$ between $1$ and the genus of $C$.



A silly question: is the number of points on a Jacobian (of a curve, over a finite field) known?
Some algorithms working in polynomial time are available, but for high values of the genus the exponent is high and the implementation is difficult. A nice survey is the paper of Gaudry and Harley


"Counting Points on Hyperelliptic Curves over Finite Fields"


Lecture Notes in Computer Science, 2000, Volume 1838/2000, 313-332,


which also contains some explicit computations on Jacobians in the case of genus $2$.


