Adjoint Jordan blocks for simple algebraic groups of type $C_{\mathrm{\ell }}$ in characteristic two

Let $G$ be a simple algebraic group over an algebraically closed field $K$ with Lie algebra $\mathfrak{g}$. For unipotent elements $u\in G$ and nilpotent elements $e\in \mathfrak{g}$, the Jordan block sizes of $\mathrm{Ad}(u)$ and $\mathrm{ad}(e)$ are known in most cases. In the cases that remain, the group $G$ is of classical type in bad characteristic, so $\mathrm{char}K=2$ and $G$ is of type $B_{\mathrm{\ell }}$, $C_{\mathrm{\ell }}$, or $D_{\mathrm{\ell }}$. In this paper, we consider the case where $G$ is of type $C_{\mathrm{\ell }}$ and $\mathrm{char}K=2$. As our main result, we determine the Jordan block sizes of $\mathrm{Ad}(u)$ and $\mathrm{ad}(e)$ for all unipotent $u\in G$ and nilpotent $e\in \mathfrak{g}$. In the case where $G$ is of adjoint type, we will also describe the Jordan block sizes on $[\mathfrak{g},\mathfrak{g}]$.