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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 $K/\mathbf{Q}_p$ isunramified, we adapt the theory of local models recently developed by Le--LeHung--Levin--Morra to study the geometry of potentially crystalline substacksin this stack. In particular, we prove the unibranch property at torus fixedpoints of local models and deduce that tamely potentially crystallinedeformation rings are domain under genericity conditions. As applications, weprove, under appropriate genericity conditions, an $\mathrm{GSp}_4$-analogue ofthe Breuil--M\'ezard conjecture for tamely potentially crystalline deformationrings, the weight part of Serre's conjecture formulated by Gee--Herzig--Savittfor global Galois representations valued in $\mathrm{GSp}_4$ satisfyingTaylor--Wiles conditions, and a modularity lifting result for tamelypotentially crystalline representations.

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