Serre weight conjectures for ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$

We prove the weight part of Serre’s conjecture for Galois representations valued in ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ that are tamely ramified with explicit genericity at places above $p$ as conjectured by Herzig–Tilouine and Gee–Herzig–Savitt. This improves on arXiv:2304.13879 where an inexplicit genericity hypothesis was required. As an application, we prove a modularity lifting theorem for ${\mathrm{G}\mathrm{S}\mathrm{p}}_4$ under similar assumptions.