This book presents an important breakthrough in arithmetic geometry. In 2014, this book's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, the author introduced the concept of “diamonds,” which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. This book shows that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. The book follows the informal style of the original Berkeley lectures, with one chapter per lecture. It explores p-adic and perfectoid spaces before laying out the newer theory of shtukas and their moduli spaces. Points of contact with other threads of the subject, including p-divisible groups, p-adic Hodge theory, and Rapoport-Zink spaces, are thoroughly explained.

We prove several results about moduli spaces of p-divisible groups such as Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level carry a natural structure as a perfectoid space, and to give a description purely in terms of padic Hodge theory of these spaces. This allows us to formulate and prove duality isomorphisms between basic Rapoport-Zink spaces at infinite level in general. Moreover, we identify the image of the period morphism, reproving results of Faltings, [Fal10]. For this, we give a general classification of p-divisible groups over O C , where C is an algebraically closed complete extension of Q p , in the spirit of Riemann's classification of complex abelian varieties. Another key ingredient is a full faithfulness result for the Dieudonné module functor for p-divisible groups over semiperfect rings (i.e. rings on which the Frobenius is surjective).

This chapter offers a second lecture on perfectoid spaces. A perfectoid Tate ring R is a complete, uniform Tate ring containing a pseudo-uniformizer. A perfectoid space is an adic space covered by affinoid adic spaces with R perfectoid. The term “affinoid perfectoid space” is ambiguous. The chapter then looks at the tilting process and the tilting equivalence. The tilting equivalence extends to the étale site of a perfectoid space. Why is it important to study perfectoid spaces? The chapter puts forward a certain philosophy which indicates that perfectoid spaces may arise even when one is only interested in classical objects.

Abstract. The cohomology of the Lubin-Tate tower is known to realize the local Langlands correspondence for GL(n) over a nonarchimedean local field. In this article we make progress towards a purely local proof of this fact. To wit, we find a family of open affinoid subsets of Lubin-Tate space at infinite level, whose cohomology realizes the local Langlands correspondence for a broad class of supercuspidals (those whose Weil parameters are induced from an unramified degree n extension). A key role is played by a certain variety X, defined over a finite field, which is "maximal" in the sense that the number of rational points of X is the largest possible among varieties with the same Betti numbers as X. The variety X is derived from a certain unipotent algebraic group, in an analogous manner as Deligne-Lusztig varieties are derived from reductive algebraic groups.

We produce an integral model for the modular curve X(N p m ) over the ring of integers of a sufficiently ramified extension of Z p whose special fiber is a semistable curve in the sense that its only singularities are normal crossings. This is done by constructing a semistable covering (in the sense of Coleman) of the supersingular part of X(N p m ), which is a union of copies of a Lubin-Tate curve. In doing so we tie together non-abelian Lubin-Tate theory to the representation-theoretic point of view afforded by Bushnell-Kutzko types.For our analysis it was essential to work with the Lubin-Tate curve not at level p m but rather at infinite level. We show that the infinitelevel Lubin-Tate space (in arbitrary dimension, over an arbitrary nonarchimedean local field) has the structure of a perfectoid space, which is in many ways simpler than the Lubin-Tate spaces of finite level.