nLab Emerton-Gee stack

Idea

The Emerton-Gee stack is the moduli stack of étale $(\varphi,\Gamma)$-modules. It is used as to study p-adic Galois modules $\rho:\Gal(\overline{K}/K)\to \GL_{d}(A)$, for $K$ a finite extension of $\mathbb{Q}_{p}$ and $A$ a p-adically complete $\mathbb{Z}_{p}$-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 $\mathbb{F}_{p}$, and is equidimensional of $[K:\mathbb{Q}_{p}]d(d-1)/2$. The irreducible components of the reduced substack are labeled by Serre weights (irreducible $\overline{\mathbb{F}}_{p}$ representations of $\GL_{d}(\mathcal{O}_{K})$).

Related concepts

• moduli stack of L-parameters

• Galois modules

• Galois representations

• p-adic Hodge theory

References

• Matthew Emerton, Toby Gee, Moduli stacks of étale $(\phi, \Gamma)$ modules and the existence of crystalline lifts (arXiv:1908.07185)

• Rebecca Bellovin, Neelima Borade, Anton Hilado, Kalyani Kansal, Heejong Lee, Brandon Levin, David Savitt, Hanneke Wiersema, Irregular loci in the Emerton-Gee stack for $\mathrm{GL}_2$ (arXiv:2309.13665)