Period rings and comparison isomorphisms in arithmetic geometry. Right now i have been focusing on padic galois representations from algebraic geometry, more specifically, integral padic hodge theory. V5a4 selected topics in algebraic geometry p adic hodge theory winter semester 201516, wednesday 16. Morrow let c be an algebraically closed complete extension of q p. Please confirm with conference organizers before making any travel plans.
While hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions in number theory. In a twotalk series titled coefficients in integral adic hodge theory via generalized ainfrepresentations and qconnections, matthew morrow and takeshi tsuji explained their recent work developing a theory of integral adic hodge theory, in the sense of earlier work. Cmi summer school notes on padic hodge theory preliminary version olivier brinon and brian conrad contents part i. Scholze short citation international mathematical union. Together with bhatt and morrow, scholze developed an integral version of padic hodge theory that establishes a relation between the torsion in betti and crystalline cohomologies. I have been interested in algebraic number theory and arithmetic geometry since 1995, when i was a graduate student in tsinghua university, china. Kiran kedlayas list of conferences in arithmetic geometry. Develops a \ p\ adic analog of fuchsian and bers uniformization theories. This course will provide an introduction to padic hodge theory, a major area of arithmetic geometry, through pdivisible groups these are also known as barsottitate groups, since the phrase pdivisible group is so ambiguous. What are padic hodge theory and padic comparison theorems good for in todays arithmetic geometry, both padic hodge theory and automorphic forms play a crucial role. On the hodge newton filtration of p divisible groups of hodge type mathematische zeitschrift 2019.
Automorphic forms and geometry of arithmetic varieties. On the way to the revolution that he launched in arithmetic geometry, scholze took. 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. After rehearsing various versions of weights, we explain some more recent applications of weights, e. Annals of mathematics studies number 207 berkeley lectures march 27, 2020 6. Multiplicative seminorms and berkovich spaces 19 5. The main method is to transfer these statements to questions about the geometry of a space parametrizing certain padic galois representations of a local galois group. Test case for an absent theory michael harris department of mathematics columbia university perfectoid prologue its not often that contemporary mathematics provides such a clearcut example of concept formation as the one i am about to present. To illustrate possible applications in arithmetic geometry we will sketch proofs of the absolute purity conjecture in etale cohomology and comparison theorems of padic hodge theory. Four years after its birth, perfectoid geometry, the theory of perfectoid spaces, is a textbook example of a progressive research program in the lakatos sense. May 29, 2017 criterion and in determine all companion forms of a given classical form i. For example, recent work on modularity of galois representations part of the langlands program, extending the resolution of the fermat problem by wiles, depends crucially on padic hodge theory. Right now i have been focusing on p adic galois representations from algebraic geometry, more specifically, integral p adic hodge theory.
Alexander paulin university of california, berkeley. A fundamental arithmetic invariant of eis the zrank of. Ostrowskis classi cation of absolute values on q 5 5. For example, recent work on modularity of galois representations part of the langlands program, extending the resolution of the fermat problem by wiles, depends crucially on p adic hodge theory. I will explain a p adic analogue of this result, using recent results of abecaro on arithmetic dmodules. Integral padic hodge theory, and qde rham cohomology peter scholze algebraic geometry salt lake city. V5a4 selected topics in algebraic geometry padic hodge theory winter semester 201516, wednesday 16. On the way, i will also introduce an arithmetic p adic category analogous to the classical ocategory appearing in classical representation theory of lie algebras and explain how objects of this category can be localized. On the hodgenewton filtration of pdivisible groups of hodge type mathematische zeitschrift 2019. Peter scholzes introduction of the new notion of perfectoid space. In the 2010s, peter scholze developed perfectoid spaces and new cohomology theories in arithmetic geometry over p adic fields with application to galois representations and certain cases of the weightmonodromy conjecture. In padic geometry, the topology is somehow adapted to the topology of the p adic numbers, and the functions take padic values. My work predominantly explores the padic and geometric aspects of the langlands program and is highly interdisciplinary, involving number theory, arithmetic geometry, algebraic geometry, padic analytic geometry, dmodule theory, padic hodge theory, motive theory and higher category theory. Workshop padic arithmetic geometry abstracts john coates, ramdorai sujatha title.
Due to the global outbreak of novel coronavirus covid19, many events listed for springsummer 2020 are likely to be postponed, cancelled, or held in an onlineonly format. V5a4 selected topics in algebraic geometry padic hodge. Techniques from padic hodge theory in commutative algebra. Cmi summer school notes on padic hodge theory preliminary version 5. Scholzes version of padic hodge theory extends to general padic rigid spaces. Masterclass stable homotopy theory and padic hodge. Gives a systematic treatment of a nonabelian example of \ p\ adic hodge theory. One interesting aspect of noncommutative iwasawa theory is that nonabelian padic lie extensions of number. The p padic hodge structure of a proper or semistably compactified padic analytic variety is essentially given by a relation between three important invariants of the given variety. Perfectoid spaces are spaces in the setting of padic geometry, and more precisely padic analytic geometry. My work predominantly explores the p adic and geometric aspects of the langlands program and is highly interdisciplinary, involving number theory, arithmetic geometry, algebraic geometry, p adic analytic geometry, dmodule theory, p adic hodge theory, motive theory and higher category theory.
Diophantine problems and padic period mappings after. The general strategy of padic hodge theory, introduced by fontaine, is to construct certain socalled period rings such as b dr, b st, b cris, and b ht which have both an action by g k and some linear algebraic structure and to consider socalled dieudonne modules. Introduction to arithmetic geometry mit mathematics. Since the notion of good reduction makes sense at without any reference to p, it is natural to ask if there is an analogue of theorem 1. He provided 2 sets of notes, on explicit reciprocity law for lubintate groups dvi, ps, pdf and on crystalline sheaves, syntomic cohomology, and padic polylogarithms dvi, ps, pdf. In recent years stable homotopy theory has seen unexpected applications to arithmetic geometry. Treats the analog of serretate theory for hyperbolic curves. Whereas arithmetic langlands studies bun gln together with its frobenius map, which gives you moduli spaces of. The relation between two fundamental methods of obtaining dimension formulas for cusp forms, the selberg trace formula and the index theorem riemannrochs theorem and the lefschetz. We give a complete description of the local geometry of this space using that it is equi.
Notes on padic hodge theory updated on april 8, 2020 teaching this winter, i am an instructor for math 115. Loosely speaking, padic hodge theory is the study of representations of padic galois groups on vector spaces with padic or characteristic pcoe. In arithmetic situations, the tools of padic hodge theory have given alternative proofs of, or analogous results to, classical hodge theory. This course will provide an introduction to p adic hodge theory, a major area of arithmetic geometry, through p divisible groups these are also known as barsottitate groups, since the phrase p divisible group is so ambiguous. This filtration is an analogue of the filtration of algebraic \k\theory by motivic cohomology. In his padic hodge theory for rigidanalytic varieties paper, scholze pursues this idea to extend the foundational results in padic hodge theory, such as faltingss work mentioned above, to the setting. One important aspect of their argument then is the interplay between the padic period map and the complex period map. Instead, lawrence and venkatesh use padic hodge theory to construct a padic period map, which encodes the variation of padic galois representations in a family of algebraic varieties.
In particular the work of matthew morrow in collaboration with bhargav bhatt and peter scholze on integral padic hodge theory was, in part, motivated by calculations of topological hochschild homology for certain arithmetically important rings. Some padic differential equations in arithmetic geometry kiran kedlaya since dworks pioneering work in the sixties, it has been known at some level that there are deep connections between complex algebraic and analytic di. Arithmetic geometry is centered around diophantine geometry, the study of rational points of algebraic varieties in more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers. But weights also appear on algebraic fundamental groups and in padic hodge theory, where they become only visible after applying the comparison functors of fontaine. Presents a systematic treatment of the moduli space of curves from the point of view of \ p\ adic galois representations. Introduction to padic hodge theory lectures by serin hong. It takes values in a mixedcharacteristic analogue of dieudonne modules, which was previously defined by fargues as a version of breuilkisin modules. In arithmetic situations, the tools of p adic hodge theory have given alternative proofs of, or analogous results to, classical hodge theory. A survey of the hodgearakelov theory of elliptic curves i. The theory has its beginnings in jeanpierre serre and john tate s study of tate modules of abelian varieties and the notion of hodge tate representation. As an application, i will show that the dmodule theoretic pushforward of an isocrystal along a proper smooth morphism is an isocrystal.
Automorphic forms and geometry of arithmetic varieties deals with the dimension formulas of various automorphic forms and the geometry of arithmetic varieties. In 2001, the proof of the local langlands conjectures for gl n was based on the geometry of certain shimura varieties. Having been introduced in the early 1980s, use of p adic hodge theory in arithmetic geometry has exploded in the last decade. The construction of the cohomology theory relies on faltingss almost purity theorem.
Number theory and representation theory seminar analytic number theory, algebraic number theory, arithmetic geometry, automorphic forms, and even some things not beginning with the letter a. Padic motivic cohomology in arithmetic request pdf. In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Schedule and notes for the 201718 seminaire godement. The goal here is to use p adic representations to study the geometry of algebraic varieties xover k. Having been introduced in the early 1980s, use of padic hodge theory in arithmetic geometry has exploded in the last decade. Integral padic hodge theory gives back classical padic hodge theory by inverting p, but it also gives rise to completely new characteristic p phenomena by reducing modulo p. My question is about p adic hodge tate theory and p adic galois representation. My question is about padic hodgetate theory and padic galois representation. 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 residual characteristic p such as q p. Apr 17, 2019 in mixed characteristic and in equal characteristic \ p \ we define a filtration on topological hochschild homology and its variants.
This filtration is an analogue of the filtration of algebraic \k\ theory by motivic cohomology. In mixed characteristic and in equal characteristic \p\ we define a filtration on topological hochschild homology and its variants. Peter scholzes lectures on padic geometry, fall 2014. Its graded pieces are related in mixed characteristic to the complex \a\omega\ constructed in our previous work, and in equal characteristic \p\ to crystalline. Hodge theory and its generalization elliptic curves. The first part of a new approach for the padic simpson correspondence, closely related to the original approach of faltings, but also inspired by the work of ogus and vologodsky on an analogue in characteristic p0.
Integral padic hodge theory, and qde rham cohomology. Tsuji gave a series of 4 lectures at usc and caltech in february, 2001 on various topics related to padic hodge theory. It is seen, retrospectively, as the right theory toward which several strands of arithmetic geometry were independently striving. The first part of a new approach for the p adic simpson correspondence, closely related to the original approach of faltings, but also inspired by the work of ogus and vologodsky on an analogue in characteristic p 0.