nLab Emerton-Gee stack

Contents

Contents

Idea

The Emerton-Gee stack is the moduli stack of étale (φ,Γ)(\varphi,\Gamma)-modules. It is used as to study p-adic Galois modules ρ:Gal(K¯/K)GL d(A)\rho:\Gal(\overline{K}/K)\to \GL_{d}(A), for KK a finite extension of p\mathbb{Q}_{p} and AA a p-adically complete p\mathbb{Z}_{p}-algebra, as it is better behaved than the naive moduli stack of these p-adic Galois modules (which sits inside of it as a substack).

Properties

The Emerton-Gee stack is a Noetherian formal algebraic stack. Its underlying reduced substack is an algebraic stack of finite type over 𝔽 p\mathbb{F}_{p}, and is equidimensional of [K: p]d(d1)/2[K:\mathbb{Q}_{p}]d(d-1)/2. The irreducible components of the reduced substack are labeled by Serre weights (irreducible 𝔽¯ p\overline{\mathbb{F}}_{p} representations of GL d(𝒪 K)\GL_{d}(\mathcal{O}_{K})).

References

Last revised on December 1, 2022 at 04:19:19. See the history of this page for a list of all contributions to it.