Let $X$ be a smooth projective curve of genus $g>1$ over an algebraically closed field $k$ of characteristic $p>0$. Let $\alpha_{p}$ be the group scheme of the kernel of $F: \mathbb{G}_{m} \rightarrow \mathbb{G}_{m}$, where $F$ denote the Frobenius. It is well-know that $H_{fppf}^{1}(X, \alpha_{p})$ is the kernel of Cartier operator on $H^{1}(X,\Omega^{1}_{X})$.

Is $H_{fppf}^{1}(X, \alpha_{p})$ a finite group? If not, what is the dim$_{k}H_{fppf}^{1}(X, \alpha_{p})$? What is the relationship between $H_{fppf}^{1}(X, \alpha_{p})$ and the $p$-rank of $X$?

Let $X$ be a smooth projective curve of genus $g>1$ over an algebraically closed field $k$ of characteristic $p>0$. Let $\alpha_{p}$ be the group scheme of the kernel of $F: \mathbb{G}_{a} \rightarrow \mathbb{G}_{a}$, where $F$ denotes the Frobenius. It is well-know that $H_{fppf}^{1}(X, \alpha_{p})$ is the kernel of Cartier operator on $H^{0}(X,\Omega^{1}_{X,d=0})$.

Is $H_{fppf}^{1}(X, \alpha_{p})$ a finite group? If not, what is the dim$_{k}H_{fppf}^{1}(X, \alpha_{p})$? What is the relationship between $H_{fppf}^{1}(X, \alpha_{p})$ and the $p$-rank of $X$?