Let $\mathbb{F}_q$ be a finite field of characteristic $p$ and let $C$ be a plane projective nonsingular curve over $\mathbb{F}_q$ , with function field $K = \mathbb{F}_q(C)$. Let $K^p$ denote the subfield of $p$-th powers and choose a separable generating transcendental element $x \in K \setminus K^p$.
Denote with $\Omega(K)$ the $K$-vector space of differentials, then every differential $\omega \in \Omega(K)$ can be written uniquely as $\omega = d \lambda + a^px^{p-1}dx$ with $\lambda, a \in K$.
The Cartier operator $\mathcal{C}: \Omega(K) \to \Omega(K)$ is defined as $\mathcal{C}(x) = a dx$.
Let $(\omega_1 , \ldots , \omega_g)$ be a basis of $\Omega^0(K)$, then there exists a $g × g$ matrix $A = (a_{i,j} )$ with coefficients in $\mathbb{F}_q$ such that $$\mathcal{C} \begin{pmatrix} \omega_1 \\ \vdots \\ \omega_g \\ \end{pmatrix} = A^{(\frac{1}{p})}\begin{pmatrix} \omega_1 \\ \vdots \\ \omega_g \\ \end{pmatrix}, \text{ where } A^{(\frac{1}{p})} \text{ denotes } (a_{i,j}^{\frac{1}{p}} ),$$ see Yu. I. Manin, 1961.
The matrix $A$ is called the Cartier–Manin matrix. My question is about a calculation of this matrix (for some $\omega_1, \ldots, \omega_g$). If $C$ is a hyperelliptic curve then there is an effective algorithm for this - that works in time $poly(g, p, \log q)$ - see Handbook of elliptic and hyperelliptic curve cryptography, pg. 412.
Is there an effective algorithm of calculation of $A$ for an arbitrary $C$?
