Berkeley Lectures on p-adic Geometry
Jared Weinstein, Peter Scholze
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
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.
Closed immersion, Formal scheme, Theory, Topological ring, Reductive group, Projective variety, Ideal (ring theory), Category of rings, Mathematical induction, Totally disconnected space, Geometry, Topological space, Cohomology, Maximal compact subgroup, Tate module, Equivalence of categories, Characterization (mathematics), Torsor (algebraic geometry), Topological group, Group (mathematics), Module (mathematics), Presheaf (category theory), Diagram (category theory), Open set, Algebraic space, Limit (category theory), Witt vector, Affine space, Linear algebraic group, Abelian variety, Compact space, Base change, Residue field, Set (mathematics), Moduli space, Existential quantification, Ring of integers, Homomorphism, Surjective function, Tensor product, Sheaf (mathematics), Zariski topology, Mathematical proof, Total order, Neighbourhood (mathematics), Crystalline cohomology, Functor, Stein factorization, Galois cohomology, Inverse system, Antiderivative, Automorphism, Facet (geometry), Pushout (category theory), Pullback (category theory), Valuation ring, Deformation theory, Conjugacy class, Connected component (graph theory), Projective module, Perfectoid, Conjecture, Transcendence degree, P-adic Hodge theory, Subset, Topology, Equivalence class, Profinite group, Isomorphism class, Fiber bundle, Morphism, Exceptional isomorphism, Field of fractions, Newton polygon, Homeomorphism, Morphism of schemes, Upper half-plane, Algebraically closed field, Spectrum of a ring, P-adic number, Divisible group, Bijection, Disjoint union, Integral element, Finitely presented, Vector bundle, Injective function, Archimedean property, C0, Special case, Theorem, Inverse limit, Analytic geometry, Quasi-projective variety, Generic point, Projective space, Subcategory, Algebraic geometry, Exterior algebra, Shimura variety