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Berkeley Lectures on p-adic Geometry presents an important
breakthrough in arithmetic geometry. In 2014, leading mathematician
Peter Scholze 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, Scholze
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. In
this book, Peter Scholze and Jared Weinstein show 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.
This 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. Berkeley Lectures
on p-adic Geometry will be a useful resource for students and
scholars working in arithmetic geometry and number theory.
Berkeley Lectures on p-adic Geometry presents an important
breakthrough in arithmetic geometry. In 2014, leading mathematician
Peter Scholze 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, Scholze
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. In
this book, Peter Scholze and Jared Weinstein show 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.
This 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. Berkeley Lectures
on p-adic Geometry will be a useful resource for students and
scholars working in arithmetic geometry and number theory.
Introduced by Peter Scholze in 2011, perfectoid spaces are a bridge
between geometry in characteristic 0 and characteristic $p$, and
have been used to solve many important problems, including cases of
the weight-monodromy conjecture and the association of Galois
representations to torsion classes in cohomology. In recognition of
the transformative impact perfectoid spaces have had on the field
of arithmetic geometry, Scholze was awarded a Fields Medal in 2018.
This book, originating from a series of lectures given at the 2017
Arizona Winter School on perfectoid spaces, provides a broad
introduction to the subject. After an introduction with insight
into the history and future of the subject by Peter Scholze, Jared
Weinstein gives a user-friendly and utilitarian account of the
theory of adic spaces. Kiran Kedlaya further develops the
foundational material, studies vector bundles on Fargues-Fontaine
curves, and introduces diamonds and shtukas over them with a view
toward the local Langlands correspondence. Bhargav Bhatt explains
the application of perfectoid spaces to comparison isomorphisms in
$p$-adic Hodge theory. Finally, Ana Caraiani explains the
application of perfectoid spaces to the construction of Galois
representations associated to torsion classes in the cohomology of
locally symmetric spaces for the general linear group. This book
will be an invaluable asset for any graduate student or researcher
interested in the theory of perfectoid spaces and their
applications.
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