A foundational account of a new construction in the p-adic
Langlands correspondence Motivated by the p-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 Zp that
parameterize etale ( , )-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 p representations of the absolute
Galois group of a p-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-Mezard conjecture. Along the way, it proves a number of
foundational results in p-adic Hodge theory that may be of
independent interest.
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