Moduli Stacks Of Etale ( , )-Modules And The Existence Of Crystalline Lifts Av Matthew Emerton, Toby Gee

<p><b>A foundational account of a new construction in the <i>p</i>-adic Langlands correspondence</b><br><br>Motivated by the <i>p</i>-adic Langlands program, this book constructs stacks that algebraize Mazur¿s formal deformation rings of local Galois representations. More precisely, it constructs Noetherian formal algebraic stacks over Spf <b>Z</b>p that parameterize ¿le (?, G)-modules; the formal completions of these stacks at points in their special fibres recover the universal deformation rings of local Galois representations. These stacks are then used to show that all mod <i>p</i> representations of the absolute Galois group of a <i>p</i>-adic local field lift to characteristic zero, and indeed admit crystalline lifts. The book explicitly describes the irreducible components of the underlying reduced substacks and discusses the relationship between the geometry of these stacks and the Breuil¿M¿rd conjecture. Along the way, it proves a number of foundational results in <i>p</i>-adi