# A silly question: is the number of points on a Jacobian (of a curve, over a finite field) known?

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 $C$.

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In the context of cryptography "compute" means "efficiently compute". The way you propose finding the number of points in the jacobian is correct but not efficient. Francesco's answer gives you some references to look at. –  Felipe Voloch Nov 11 '10 at 13:23
which also contains some explicit computations on Jacobians in the case of genus $2$.