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

Abstract: We give a purely local proof of the explicit Local Langlands Correspondence for $\mathrm{GSp}_4$ and $\mathrm{Sp}_4$. 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.