In 1994 he was an invited speaker with talk nonarchimedean period domains at the icm in zurich. Rotman, we face to the following lemma in chapter 10. On period spaces for pdivisible groups request pdf. In another reference, an introduction to the theory of groups by j. For p p a prime number a group is p pdivisible if the above formula holds for all n n of the form p k pk for k. We give a new dieudonn e theory which associates to a formal pdivisible group xover an excellent padic ring ran object of linear algebra called. Some papers and preprints of chingli chai the more recent ones are preprints. Using the concept of rigidanalytic period maps the relation of padic period domains to moduli space of pdivisible groups is investigated. Hoaran wang, moduli spaces of pdivisible groups and period morphisms, masters thesis, 2009, pdf. The space m is a formal scheme representing a moduli functor of pdivisible groups. The padic uniformization of shimura varieties 273 bibliography 317 index 323.
Moduli spaces of pdivisible groups and period morphisms. Our main goal is to prove that rapoportzink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in. Letwl be its ring of witt vector andk0k0l wlqits fraction field. Tate modules, pdivisible groups, and the fundamental group. The analogous notion at the prime pis that of dieudonn e modules. Introduction padic symmetric domains quasiisogenies of pdivisible groups moduli spaces of pdivisible groups appendix. On a conjecture of rapoport and zink internet archive. The multiplication by p on a pdivisible group is obviously an isogeny. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian. It follows that for pdivisible groups, the group hom sg. Are there anypdivisiblegroups over z other than products ofpowers ofgmp and ofqplzp1 2. In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n.
Remark there is also an abstract notion of p p divisible group in terms of group schemes. In this talk the algebraic object will be a finite group scheme or a pdivisible group and the geometric objects considered are abelian varieties. Work on these articles have been supported by the national science foundation since 1990, including the following grants. Pel type rapoportzink space deformation space of pdivisible groups with. The serretate theorem provides a converse in the case of in. On the hodgenewton filtration for pdivisible groups with. The integer h whenever it is a constant is called the height of the pdivisible group g. On period spaces for pdivisible groups sciencedirect.
Period spaces for pdivisible groups am141, volume 141. Let r be a complete noetherian local ring, m its maximal ideal. Some papers and preprints of chingli chai penn math. Normal forms of lattice chains the formal hecke correspondences the period morphism and the rigidanalytic coverings padic uniformization of shimura varieties. We determine the image of these period morphisms, thereby contributing to a question of grothendieck. In their book rapoport and zink constructed rigid analytic period spaces for fontaines filtered isocrystals, and period morphisms from moduli spaces of pdivisible groups to some of these. Available formats pdf please select a format to send. Michael rapoport is the son of pediatrician ingeborg rapoport and biochemist samuel mitja rapoport, and brother of biochemist tom rapoport. We prove several results about pdivisible groups and rapoportzink spaces.
We determine the image of these period morphisms, thereby answering a question of grothendieck. Using the concept of rigidanalytic period maps the relation ofpadic period domains to moduli space ofpdivisible groups is investigated. This article is in its final form and can be cited using the date of online publication and the doi. Relations with formal lie groups in this section we assume r complete, noetherian, local, with residue field k ofcharacteristicp o. If this holds only for those n which are powers of a fixed prime p, we obtain the notion of pdivisible group. In this post we discuss the basic theory of pdivisible groups, their relationship to formal groups, and the serretate theorem. Dieudonn e modules and pdivisible groups brian lawrence september 26, 2014 the notion of adic tate modules, for primes away from the characteristic of the ground eld, is incredibly useful. The purpose of this course was to carefully lay the foundations for. In our talk we explained our results from 5 on the image of the rapoportzink period morphism. The theory of pdivisible groups plays an important role in arithmetic.
A quasiisogeny of pdivisible groups from g to g0 is a global section of. Our main goal is to prove that rapoportzink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in terms of padic hodge theory. The most exciting thing discused will be the relation between connected pdivisible groups and divisible formal lie groups over r. We give examples showing that only in rare cases the image is all of the rapoportzink period space. We describe families of abelian varieties, of finite group schemes, of pdivisible groups. If x is an abelian variety over the complex numbers, the associated analytic manifold can be described as a complex torus v. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
We give definitions, properties, examples, and open problems. We determine the image of these period morphisms, thereby contributing to a question of. Notes on pdivisible groups march 24, 2006 this is a note for the talk in stage in mit. Statement of purpose the aim here is simply to provide some details to some of the proofs in tates paper t. Period spaces for p divisible groups am141 export a ris file for endnote, procite, reference manager, zotero, mendeley export a text file for bibtex note. Rapoports students include maria heepaltiner, werner baer, peter scholze, eva viehmann. A group g is called divisible if every equation of the form nx g, with n. T is a morphism, a deformation of a s is an abelian scheme a0 t together with isomorphisms a 0 t t s. We study the geometry of the special fibers of certain shimura curves and give a direct proof of globaltolocal jacquetlanglands compatibility by cerednikdrinfeld uniformizations theorem. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.
Jared weinstein, the geometry of lubitate spaces, lecture 1. In their book rapoport and zink constructed rigid analytic period spaces for fontaines filtered isocrystals, and period morphisms from moduli spaces of pdivisible groups to some of these period spaces. We study the global structure of moduli spaces of quasiisogenies of polarized pdivisible groups introduced by rapoport and zink. In this monographpadic period domains are associated to arbitrary reductive groups. This write up is devoted to some basic but important aspects of the structure theory of pdivisible groups. This allows us to formulate and prove duality isomorphisms between basic rapoportzink spaces at infinite level in general. The moduli spaces obtained in this way are formal schemes over the. April 25, 2008 abstract in their book rapoport and zink constructed rigid analytic period spaces for fontaines. Displays and formal p divisible groups springerlink.
Historically, results and conjectures surrounding pdivisible groups were the main stimulus for the development of padic hodge theory, and they continue to be relevant in modern research. In addition, nonarchimedean uniformization theorems for general shimura varieties are established. Other readers will always be interested in your opinion of the books youve read. A group r is reduced if it has no nonzero divisible subgroups. For both abelian schemes and pdivisible groups, we have the notion of deformations.
The period morphism and the rigidanalytic coverings. I will explain why moduli spaces of pdivisible groups become perfectoid spaces. Using the concept of rigidanalytic period maps the relation of p adic period domains to moduli space of p divisible groups is investigated. Period spaces for pdivisible groups am141 by michael rapoport, thomas zink pdf, epub ebook d0wnl0ad in this monograph p adic period domains are associated to arbitrary reductive groups. It can be shown that a nite group is nilpotent if and only if it possesses a central series. Explain sketch how the notion of pdivisible group is related to the notion of formal group. Using the corresponding results for nonpolarized pdivisible groups from a previous paper, we determine their dimensions and their sets of connected components and of irreducible components. This construction is based on the moduli theory of pdivisible groups of a fixed isogeny type. On period spaces for pdivisible groups urs hartl in our talk we.
For ourpresentpurposes, an ndimensional formal lie group rover r can bedefined as a suitable. Annals of mathematics, 161 2005, 10211036 minimal pdivisible groups by frans oort introduction a pdivisible group x can be seen as a tower of building blocks, each of which is isomorphic to the same. The space of level 0, and the period morphism 44 6. The display of a formal pdivisible group by thomas zink abstract. Motivation the needdesire for pdivisible groups when one studies the arithmetic of elliptic curves or, more generally, abelian varieties there is an object of paramount importance which rears its head in almost all aspects of the. Note that if the equation nx g has solutions then we also say that n divides g, and we write ng.
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