Introduction to arithmetic geometry mit mathematics. In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Alexander paulin university of california, berkeley. 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. Please confirm with conference organizers before making any travel plans. 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. Tsuji gave a series of 4 lectures at usc and caltech in february, 2001 on various topics related to padic hodge theory. Masterclass stable homotopy theory and padic hodge. 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. V5a4 selected topics in algebraic geometry padic hodge theory winter semester 201516, wednesday 16.
Right now i have been focusing on p adic galois representations from algebraic geometry, more specifically, integral p adic hodge theory. In arithmetic situations, the tools of padic hodge theory have given alternative proofs of, or analogous results to, classical hodge theory. Automorphic forms and geometry of arithmetic varieties. Peter scholzes introduction of the new notion of perfectoid space.
Morrow let c be an algebraically closed complete extension of q p. 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. This filtration is an analogue of the filtration of algebraic \k\theory by motivic cohomology. Perfectoid spaces are spaces in the setting of padic geometry, and more precisely padic analytic geometry. 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. 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.
Scholze short citation international mathematical union. V5a4 selected topics in algebraic geometry p adic hodge theory winter semester 201516, wednesday 16. 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. 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. 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.
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. Cmi summer school notes on padic hodge theory preliminary version olivier brinon and brian conrad contents part i. Presents a systematic treatment of the moduli space of curves from the point of view of \ p\ adic galois representations. My question is about p adic hodge tate theory and p adic galois representation. On the hodgenewton filtration of pdivisible groups of hodge type mathematische zeitschrift 2019. The construction of the cohomology theory relies on faltingss almost purity theorem. Loosely speaking, padic hodge theory is the study of representations of padic galois groups on vector spaces with padic or characteristic pcoe. As an application, i will show that the dmodule theoretic pushforward of an isocrystal along a proper smooth morphism is an isocrystal. Diophantine problems and padic period mappings after.
Having been introduced in the early 1980s, use of p adic hodge theory in arithmetic geometry has exploded in the last decade. Ostrowskis classi cation of absolute values on q 5 5. While hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions in number theory. Padic motivic cohomology in arithmetic request pdf. Annals of mathematics studies number 207 berkeley lectures march 27, 2020 6. 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. Whereas arithmetic langlands studies bun gln together with its frobenius map, which gives you moduli spaces of. 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.
Cmi summer school notes on padic hodge theory preliminary version 5. Workshop padic arithmetic geometry abstracts john coates, ramdorai sujatha title. Kiran kedlayas list of conferences in arithmetic geometry. In arithmetic situations, the tools of p adic hodge theory have given alternative proofs of, or analogous results to, classical hodge theory. I will explain a p adic analogue of this result, using recent results of abecaro on arithmetic dmodules. 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. 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. 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. The goal here is to use p adic representations to study the geometry of algebraic varieties xover k. Notes on padic hodge theory updated on april 8, 2020 teaching this winter, i am an instructor for math 115. 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. 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.
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. Techniques from padic hodge theory in commutative algebra. 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. 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. Scholzes version of padic hodge theory extends to general padic rigid spaces. Introduction to padic hodge theory lectures by serin hong. Develops a \ p\ adic analog of fuchsian and bers uniformization theories. 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. Treats the analog of serretate theory for hyperbolic curves. 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. 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. 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.
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. 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. In padic geometry, the topology is somehow adapted to the topology of the p adic numbers, and the functions take padic values. 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. 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. Integral padic hodge theory, and qde rham cohomology peter scholze algebraic geometry salt lake city. Right now i have been focusing on padic galois representations from algebraic geometry, more specifically, integral padic hodge theory. May 29, 2017 criterion and in determine all companion forms of a given classical form i. My question is about padic hodgetate theory and padic galois representation. 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. One interesting aspect of noncommutative iwasawa theory is that nonabelian padic lie extensions of number.
Schedule and notes for the 201718 seminaire godement. After rehearsing various versions of weights, we explain some more recent applications of weights, e. 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. On the way to the revolution that he launched in arithmetic geometry, scholze took. 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. Multiplicative seminorms and berkovich spaces 19 5. V5a4 selected topics in algebraic geometry padic hodge. On the hodge newton filtration of p divisible groups of hodge type mathematische zeitschrift 2019. It takes values in a mixedcharacteristic analogue of dieudonne modules, which was previously defined by fargues as a version of breuilkisin modules. Automorphic forms and geometry of arithmetic varieties deals with the dimension formulas of various automorphic forms and the geometry of arithmetic varieties. We give a complete description of the local geometry of this space using that it is equi. 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. Peter scholzes lectures on padic geometry, fall 2014. 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.
In recent years stable homotopy theory has seen unexpected applications to arithmetic geometry. It is seen, retrospectively, as the right theory toward which several strands of arithmetic geometry were independently striving. Gives a systematic treatment of a nonabelian example of \ p\ adic hodge theory. A fundamental arithmetic invariant of eis the zrank of. Period rings and comparison isomorphisms in arithmetic geometry. Having been introduced in the early 1980s, use of padic hodge theory in arithmetic geometry has exploded in the last decade. 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. Integral padic hodge theory, and qde rham cohomology. Hodge theory and its generalization elliptic curves.
I have been interested in algebraic number theory and arithmetic geometry since 1995, when i was a graduate student in tsinghua university, china. Apr 17, 2019 in mixed characteristic and in equal characteristic \ p \ we define a filtration on topological hochschild homology and its variants. A survey of the hodgearakelov theory of elliptic curves i. In 2001, the proof of the local langlands conjectures for gl n was based on the geometry of certain shimura varieties. One important aspect of their argument then is the interplay between the padic period map and the complex period map. Notes on p adic hodge theory updated on april 8, 2020 teaching this winter, i am an instructor for math 115. 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.