Let $k$ be an algebraically closed field of characteristic $p >0$, $X$ a smooth projective curve of genus $g \geqslant 2$ over $k$, and $\mathscr{E}$ a vector bundle (or locally free sheaf) on $X$ of rank $n$.
We define the degree of $\mathscr{E}$ to be the degree of the divisor associated to the line bundle $\det (\mathscr{E})$, which is the $n$-th exterior of $\mathscr{E}$.
And let $F:X \to X$ be the absolute (or relative) Frobenius morphism of $X$ ( $F_{\#}: \mathscr{O}_X \to \mathscr{O}_X$ is the $p$-th power of the structure sheaf). We denote the push-forward (resp. pull-back) of $\mathscr{E}$ by $F_*\mathscr{E}$ (resp.$F^*\mathscr{E}$).
My Question: What is the rank and degree of $F_*\mathscr{E}$ and $F^*\mathscr{E}$?
I think $\deg (F_*\mathscr{E}) = \deg(\mathscr{E}) +\operatorname{rank}(\mathscr{E})\cdot(p-1)(g-1)$ and $\deg(F^*\mathscr{E}) = p\cdot\deg(\mathscr{E})$. Is that right?
About the rank of $F_*\mathscr{E}$ , I have seen someone said that $\operatorname{rank}(F_*\mathscr{E}) = pn$ , but I don't know how to prove it. And I have no idea about $\operatorname{rank}(F^*\mathscr{E})$.
I appreciate any kind of help!
Sorry if it is an obvious thing, please just give me a reference paper. Thx!