The explicit Local Langlands Correspondence for ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$, ${\mathrm{S}\mathrm{p}}_4$ and stability (with an application to Modularity Lifting)

We give a purely local proof of the explicit Local Langlands Correspondence for ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ and ${\mathrm{S}\mathrm{p}}_4$. Moreover, we give a unique characterization in terms of stability of $L$-packets and other properties. Finally, in the appendix, we give an application of our explicit local Langlands correspondence to modularity lifting.