Emerton--Gee stacks, Serre weights, and Breuil--M\'ezard conjectures for $\mathrm{GSp}_4$

We construct a moduli stack of rank 4 symplectic projective \'etale $(\varphi,\Gamma)$-modules and prove its geometric properties for any prime $p>2$ and finite extension $K/\mathbf{Q}_p$. When $K/\mathbf{Q}_p$ is unramified, we adapt the theory of local models recently developed by Le--Le Hung--Levin--Morra to study the geometry of potentially crystalline substacks in this stack. In particular, we prove the unibranch property at torus fixed points of local models and deduce that tamely potentially crystalline deformation rings are domain under genericity conditions. As applications, we prove, under appropriate genericity conditions, an $\mathrm{GSp}_4$-analogue of the Breuil--M\'ezard conjecture for tamely potentially crystalline deformation rings, the weight part of Serre's conjecture formulated by Gee--Herzig--Savitt for global Galois representations valued in $\mathrm{GSp}_4$ satisfying Taylor--Wiles conditions, and a modularity lifting result for tamely potentially crystalline representations.