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We give a purely local proof of the explicit Local Langlands Correspondencefor $\mathrm{GSp}_4$ and $\mathrm{Sp}_4$. Moreover, we give a uniquecharacterization in terms of stability of $L$-packets and other properties.Finally, in the appendix, we give an application of our explicit localLanglands correspondence to modularity lifting.

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