Let abe an abelian variety over a eld kwith characteristic p0 and dimension g0. Using the concept of rigidanalytic period maps the relation of padic period domains to moduli space of pdivisible groups is investigated. Cm pdivisible groups over finite fields by xinyun sun a dissertation submitted in partial ful llment of the requirements for the degree of doctor of philosophy mathematics in the university of michigan 2011 doctoral committee. 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. Then the kgroups ki a are uniquely pdivisible for i 0 see 4 or 167. On period spaces for pdivisible groups sciencedirect. Nontrivial divisible groups are not finitely generated.
Hn are morphisms of group schemes and compatible with the transition maps. The display of a formal p divisible group by thomas zink abstract. Fp, the moduli space of polarized abelian varieties of dimension g in positive. The usefulness of in these constructions is based on the fact that reduces questions about uncountable subsets of. Rapoportzink spaces are deformation spaces for pdivisible groups with additional. Tag 910 pdivisible groups pdivisible groups pick a primepand work overspfzp. Tate 1967 defined a pdivisible group of height h over a scheme s to be an inductive system of groups g n for n. The space of level 0, and the period morphism 44 6. The purpose of this course was to carefully lay the foundations for. It seems to me that they can arise as summands in the p divl group of an abelian variety, just not as the whole p divl group. Consider all the strings of length mathpmath, where each position of the string can be occupied by one of mathamath distinct alphabets. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups.
An abelian group is divisible if and only if it is p divisible for every prime p. Jul 26, 2012 jacob lurie, harvard university abstract. On the other hand, reading this book is really indispensable if you want to master p. On the other hand, reading this book is really indispensable if you want to master p divisible groups. We give a new dieudonn e theory which associates to a formal p divisible group xover an excellent p adic ring ran object of linear algebra called. The system is then a divisible group of height and is called the serre dual to the divisible group. To each barsottitate group g, we can associate a dieudonn.
X 2pdoes in general not imply that x 1 and x 2 are isomorphic. Moduli spaces of pdivisible groups and period morphisms. The smooth locus in infinitelevel rapoportzink spaces. H where g and h are pdivisible groups on s is a morphism of pdivisible groups if f is a morphism of f.
Are there anypdivisiblegroups over z other than products ofpowers ofgmp and ofqplzp1 2. The first is finiteness of padic etale cohomology, corresponding to finiteness of singular. A quasiisogeny of pdivisible groups from g to g0 is a global section of. April 25, 2008 abstract in their book rapoport and zink constructed rigid analytic period spaces for fontaines.
Jared weinstein, the geometry of lubitate spaces, lecture 1. On torsion in the cohomology of locally symmetric varieties peter sholze free download as pdf file. Hoaran wang, moduli spaces of pdivisible groups and period morphisms, masters thesis, 2009, pdf. Consider all the strings of length math p math, where each position of the string can be occupied by one of mathamath distinct alphabets. If p is divisible by 3 and q is divisible by 4, then the product of these numbers must have factors of 3 and 4, which includes any number that is a factor of 12. The restriction of the functor bt to the category of displays is faithful. The serretate theorem provides a converse in the case of in. There is a very beautiful combinatorial argument that proves the above proposition. Every abelian group can be embedded in a divisible group.
This operation has been used in constructions of various mathematical objects of complex behavior such as groups, geometries, and banach spaces 20, 21, 7, 8. A 3 b 4 c 6 d 9 e 12 answer by rapaljer4671 show source. The moduli spaces obtained in this way are formal schemes over the. It seems to me that they can arise as summands in the pdivl group of an abelian variety, just not as the whole pdivl group. As in formal groups, so, too, in divisible groups, the concept of a dieudonne module, which plays an important role in the deformation theory of divisible groups, may be introduced.
Remark there is also an abstract notion of p p divisible group in terms of group schemes. Families of pdivisible groups with constant newton polygon oort, f zink, t. The local structure of algebraic ktheory pdf free download. One of the main theorems of representation theory asserts that the construction which assigns to each representation of g its. For ourpresentpurposes, an ndimensional formal lie group rover r can bedefined as a suitable homomorphism of. In a nutshell, a pdivisible group x s over a base scheme s in equi.
What is the probability of rolling a number that is divisible by 2 or 3. If a is a free associative ring, then the canonical. We prove that there exists an isogeny to x y such that y admits a slope filltration. This is a set of notes for a course we gave in the second week of august in the 2006 cmi summer school at go. The theory of pdivisible groups plays an important role in arithmetic. Let r be a complete noetherian local ring, m its maximal ideal. Associate professor kartik prasanna, cochair professor brian d.
In this monograph padic period domains are associated to arbitrary reductive groups. Adic pdivisible groups, i 2 od the connected component of g n over each strict henselization of r n is a multiplicative group i. Loop spaces, pdivisible groups, and character theory university of notre dame. These methods give rise to geometric structures, when we keep these invariants fixed. For both abelian schemes and pdivisible groups, we have the notion of deformations. Classifying spaces and spectral sequences render a chessboard from a pgn file should i ask my company to pay my internet bill if circumstances and the company strongly recommend remote work. Mar 02, 2018 in this talk the algebraic object will be a finite group scheme or a p divisible group and the geometric objects considered are abelian varieties. Zink, period spaces for pdivisible groups, annals of mathematics. Patrick walls dieudonn e modules and p divisible groups. September24,2018 abstract in their book rapoport and zink constructed rigid analytic period spaces for fontaines. Relations with formal lie groups in this section we assume r complete, noetherian, local, with residue field k ofcharacteristicp o. 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.
We study truncated barsottitate groups of level one bt1 and their extensions to pdivisible groups. We determine the image of these period morphisms, thereby contributing to a question of grothendieck. This has the implication that we will be working withpcomplete spectra. In the paper groupes pdivisible sur les corps locaux fon2. Applications to pdivisible groups and finite group schemes. We give definitions, properties, examples, and open problems. What is a proof that apa is divisible by p where p is. The general structure of these results is the following.
This construction is based on the moduli theory of pdivisible groups of a fixed isogeny type. We prove several results about pdivisible groups and rapoportzink spaces. Lectures on pdivisible groups lecture notes in mathematics. Further, every abelian group can be embedded in a divisible group as an essential subgroup in a unique way. Patrick walls dieudonn e modules and pdivisible groups. Pis a formal pdivisible group of height equal to rank rp. From the point of view of padic hodge theory, this is a foundational paper and within this setting, much of the. On torsion in the cohomology of locally symmetric varieties. Let x be a pdivisible group with constant newton polygon over a normal noetherian scheme s.
Remark what is additionally interesting about example is that it provides an. Note that if the equation nx g has solutions then we also say that n divides g, and we write ng. Windows for displays of pdivisible groups thomas zink introduction the starting point of this work was the classi cation of pdivisible groups over a discrete valuation ring of characteristic 0 with perfect residue eld of characteristic p 3 obtained by c. Firstly we show that any bt1 contains a certain minimal bt1 as a nonzero subgroup scheme. The multiplication by p on a pdivisible group is obviously an isogeny. A group g is called divisible if every equation of the form nx g, with n. Proceedings of a conference on local fields, springer, 1967. Let x be a p divisible group with constant newton polygon over a normal noetherian scheme s. 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. What is a proof that apa is divisible by p where p is a. The basics of the theory are welldeveloped in the first few chapters, but the last 2 chapters are too short. The most exciting thing discused will be the relation between connected pdivisible groups and divisible formal lie groups over r.
Therefore all the pdivisible groups on a base scheme s form a category denoted. It is fully faithful, if the ideal of nilpotent elements in ris a nilpotent ideal. 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. It follows that for pdivisible groups, the group hom sg. Theorem the witt group scheme w n of length n is an injective cogenerator in a n. The space m is a formal scheme representing a moduli functor of pdivisible groups. Families of pdivisible groups with constant newton polygon. On period spaces for pdivisible groups request pdf. Cm p divisible groups over finite fields by xinyun sun a dissertation submitted in partial ful llment of the requirements for the degree of doctor of philosophy mathematics in the university of michigan 2011 doctoral committee. Windows for displays of p divisible groups thomas zink introduction the starting point of this work was the classi cation of p divisible groups over a discrete valuation ring of characteristic 0 with perfect residue eld of.
Statement of main results the goal of this paper is to present a theory of rpointed stable curves of genus g over padic schemes for p odd, which, on the one hand, generalizes the serretate theory of ordinary elliptic curves to the hyperbolic case i. Moduli of abelian varieties and pdivisible groups chingli chai and frans oort abstract. T is a morphism, a deformation of a s is an abelian scheme a0 t together with isomorphisms a 0 t t s. Notes on pdivisible groups march 24, 2006 this is a note for the talk in stage in mit. We prove several results about moduli spaces of pdivisible groups such as.
Part ii brandon levin this talk discusses the main results of tates paper pdivisible groups 6. 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 theory of commutative formal groups over elds of nite characteristic. K on the completion c of the algebraic closure of a local field k of characteristic 0. This write up is devoted to some basic but important aspects of the structure theory of pdivisible groups. Period spaces for pdivisible groups am141 by michael rapoport, thomas zink pdf, epub ebook d0wnl0ad. We give examples showing that only in rare cases the image is all of the rapoportzink. 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. Oct 19, 2014 there is a very beautiful combinatorial argument that proves the above proposition. Combine the discussion above, to any pair, b where is a one parameter subgroup.
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