Nt 9 apr 2020 integral p adic hodge theory of formal schemes in low ramification yu min abstract. Local asphericity of smooth morphisms of schemes 544 7. Cmi summer school notes on padic hodge theory preliminary version olivier brinon and brian conrad contents part i. Nt apr 20 moduli of p divisible groups peter scholze and jared weinstein abstract. A rst glimpse of p adic hodge theory our goal in this section is to give a rough idea of what p adic hodge theory is about. Starting with the work of fontaine, who introduced many of the basic notions in the domain, it took many years to understand the exact definition of a p adic hodge structure. Further developments were inspired by properties of padic galois representations arising from the etale cohomology of varieties. Pdf we prove an analogue of schmid nilpotent orbit theorems in p adic hodge theory, by employing standard facts from p adic fourier theory of. There is a classical relation between the padic absolute value of the eigenvalues of frobenius on crystalline cohomology and hodge numbers, for a variety in characteristic p. By nature, p adic hodge theory has two sides of the story, namely the arithmetic side and the geometric side.
Consider the variety xa1 0 over the eld c of complex numbers. More generally, such a decomposition holds for compact k. It takes values in a mixedcharacteristic analogue of dieudonn e mod. We prove that for any proper smooth formal scheme x ove. It gives rise to p torsion phenomena which makes the theory very complicated.
Jun 26, 2018 in particular, we can express padic etale ktheory in terms of the cohomology theories of integral padic hodge theory, such as the prismatic cohomology discussed in bhatts talk. Inductive construction of the p adic zeta functions for noncommutative p extensions of. One of the goals of padic hodge theory is to prove similar results in the case where cis replaced instead by a padic. Introduction to paper and classical p adic hodge theory. We mention two applications of the plmt in p adic hodge theory. Pdf we prove an analogue of schmid nilpotent orbit theorems in padic hodge theory, by employing standard facts from padic fourier theory of. In particular, to each adic space over q p is associated a category of relative. We prove several results about moduli spaces of p divisible groups. Introduction to p adic hodge theory lectures by serin hong. An overview of p adic hodge theory, following bhattmorrowscholze ii sanath devalapurkar letx beasmoothpropervarietyovera.
This filtration is an analogue of the filtration of algebraic \k\ theory by motivic cohomology. Integral padic hodge theory, and qde rham cohomology. In arithmetic situations, the tools of p adic hodge theory have given alternative proofs of, or analogous results to, classical hodge theory. Alexander beilinson, padic hodge theory, lectures from. Introduction, the complex case we let kbe a eld, and xbe a smooth kscheme. Cmi summer school notes on p adic hodge theory preliminary version 5 than ae,z. We will brie y motivate and describe each side of the story, and discuss how the two sides are related. Fontaines formalism robert sayer october 30, 2014 0. We build upon faltings theory of almost etale extensions, ampli. Iwasawa theory and padic hodge theory by kazuya kato the aim of this paper is to formulate the iwasawa main conjecture for varieties or motives over arbitrary number fields.
Local acyclicity of smooth morphisms of log schemes 576 8. Period morphisms and variation of padic hodge structure. This question was rst answered by grothendieck using p divisible groups, and his answer can be put in a more useful form by means of some deep results in p adic hodge theory. Notes by aleksander horawa these are notes from math 679 taught by serin hong in winter 2020, latexed by aleksander horawa who is the only person responsible for any mistakes that may be found in them. Rather than stemming from a wellthoughtout plan, this part is the outcome of a haphazard process, lumbering between alternating phases of. Mo g c corresponding to a curve c, the fundamental group. Integral padic hodge theory could be today defined as the study of galois. Luc illusie, around the poincare lemma, after beilinson preliminary notes, 20, pdf. An example for i p adic hodge theory based on systematic use of witt vector constructions and nonarchimedean analytic geometry in the style of berkovich. Let x be a connected a noid rigidanalytic variety over k. Let e be an elliptic curve over a number field f, and fix an algebraic. Organized by bhargav bhatt university of michigan and martin olsson uc berkeley, the symposium will bring together experts to explore recent developments in adic hodge theory and understand the emerging relationship of adic hodge theory with other subjects in mathematics.
In mathematics, padic hodge theory is a theory that provides a way to classify and study padic galois representations of characteristic 0 local fields with. We construct a new cohomology theory for proper smooth formal schemes over the ring of integers of cp. Mar 24, 2008 towards nonabelian padic hodge theory 3 a point p. The padic hodge theory of semistable galois representations.
One can ask whether a similar hodgelike decomposition exists for the etale cohomology with values. Pro etale site and rational p adic hodge theory after scholze. Integral p adic hodge theory bhargav bhatt, matthew morrow, and peter scholze abstract. Integral p adic hodge theory 3 rank 2gon which g k acts continuously for the p adic topology. The hodge decomposition is proved using transcendental methods such as the use of a k. While hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions in number theory. Apr 17, 2019 in mixed characteristic and in equal characteristic \ p \ we define a filtration on topological hochschild homology and its variants. One important aspect of their argument then is the interplay between the p adic period map and the complex period map. Below you will find information about where the seminar meets, a schedule of talks for the seminar and relevant reading for each talk. Period morphisms and variation of padic hodge structure preliminary draft david hansen.