Remark what is additionally interesting about example is that it provides an. Pis a formal pdivisible group of height equal to rank rp. On the other hand, reading this book is really indispensable if you want to master p divisible groups. Zink, period spaces for pdivisible groups, annals of. Moduli of abelian varieties and pdivisible groups penn math. For both abelian schemes and pdivisible groups, we have the notion of deformations. 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. This question was rst answered by grothendieck using pdivisible groups, and his answer can be put in a more useful form by means of some deep results in padic hodge theory. It follows that for pdivisible groups, the group homsg, g is a torsion free zpmodule. If x is an abelian variety over the complex numbers, the associated analytic manifold can be described as a complex torus v. Period spaces for pdivisible groups am141, volume 141. The restriction of the functor bt to the category of displays is faithful. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Applications to pdivisible groups and finite group schemes.
You can specify the type of files you want, for your gadget. The projectivity of sh and the fact that gzg is anisotropic over qare related to each other and essential in our argument. Some papers and preprints of chingli chai penn math. Jul 26, 2012 jacob lurie, harvard university abstract. Hn are morphisms of group schemes and compatible with the transition maps. The basics of the theory are welldeveloped in the first few chapters, but the last 2 chapters are too short. Galois representations arising from some compact shimura varieties by sug woo shin abstract our aim is to establish some new cases of the global langlands correspondence for gl m. 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. This has the implication that we will be working withpcomplete spectra.
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. One of the main theorems of representation theory asserts that the construction which assigns to each representation of g its. Msri workshop schedules an overview of the theory of. K on the completion c of the algebraic closure of a local field k of characteristic 0. Nov 14, 2007 over padic nagata rings, formal pdivisible groups are classified by nilpotent displays according to t. 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.
Tag 910 pdivisible groups pdivisible groups pick a primepand work overspfzp. Lectures on pdivisible groups lecture notes in mathematics. The display of a formal p divisible group by thomas zink abstract. On a conjecture of rapoport and zink internet archive. This write up is devoted to some basic but important aspects of the structure theory of pdivisible groups. Displays and formal p divisible groups springerlink.
The period morphism and the rigidanalytic coverings. It follows that for pdivisible groups, the group hom sg. 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. Patrick walls dieudonn e modules and pdivisible groups. Let a p denote the pdivisible group of ptorsion points of a.
Relations with formal lie groups in this section we assume r complete, noetherian, local, with residue field k ofcharacteristicp o. I was recommended this book by a dear friend of mine. Hoaran wang, moduli spaces of pdivisible groups and period morphisms, masters thesis, 2009, pdf. We give examples showing that only in rare cases the image is all of the rapoportzink period space.
Moduli of abelian varieties and pdivisible groups chingli chai and frans oort abstract. A quasiisogeny of pdivisible groups from g to g0 is a global section of. We prove several results about moduli spaces of pdivisible groups such as rapoportzink spaces. Moduli spaces of pdivisible groups and period morphisms. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Etale and connected groups in topology, we know that for reasonable connected spaces sand a choice of base point 2s. Notes on pdivisible groups march 24, 2006 this is a note for the talk in stage in mit. T is a morphism, a deformation of a s is an abelian scheme a0 t together with isomorphisms a 0 t t s. Firstly we show that any bt1 contains a certain minimal bt1 as a nonzero subgroup scheme. Period spaces for pdivisible groups am141 michael rapoport, thomas zink. On the other hand, reading this book is really indispensable if you want to master pdivisible groups. The theory of commutative formal groups over elds of nite characteristic. Fp, the moduli space of polarized abelian varieties of dimension g in positive.
The space of level 0, and the period morphism 44 6. For ourpresentpurposes, an ndimensional formal lie group rover r can bedefined as a suitable homomorphism of. Every abelian group can be embedded in a divisible group. Moreover, we identify the image of the period morphism, reproving results of faltings. The multiplication by p on a pdivisible group is obviously an isogeny. 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. Our main goal is to prove that rapoportzink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces. In this article we treat the essential case where the barsottitate group has exactly two slopes. 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. I will explain why moduli spaces of pdivisible groups become perfectoid spaces at infinite level. Letwl be its ring of witt vector andk0k0l wlqits fraction field.
The moduli spaces obtained in this way are formal schemes over the. One signi cant consequence of this is that it yields a simple description of pdivisible groups over the ring. Tate modules, pdivisible groups, and the fundamental group. Tate 1967 defined a pdivisible group of height h over a scheme s to be an inductive system of groups g n for n. 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. Jared weinstein, the geometry of lubitate spaces, lecture 1. Adic pdivisible groups, i 2 od the connected component of g n over each strict henselization of r n is a multiplicative group i.
Let abe an abelian variety over a eld kwith characteristic p0 and dimension g0. Sustained pdivisible groups and a foliation on moduli spaces of abelian varieties. X 2pdoes in general not imply that x 1 and x 2 are isomorphic. Using the concept of rigidanalytic period maps the relation of padic period domains to moduli space of pdivisible groups is investigated. Our main goal is to prove that rapoportzink spaces at infinite level are naturally perfectoid spaces, and to give a. Work on these articles have been supported by the national science foundation since 1990, including the following grants. Over padic nagata rings, formal pdivisible groups are classified by nilpotent displays according to t.
The proof uses the grothendieckillusie deformation theory of truncated pdivisible groups. Theorem the witt group scheme w n of length n is an injective cogenerator in a n. From the point of view of padic hodge theory, this is a foundational paper and within this setting, much of the. An abelian group is divisible if and only if it is p divisible for every prime p. 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. September24,2018 abstract in their book rapoport and zink constructed rigid analytic period spaces for fontaines. Part ii brandon levin this talk discusses the main results of tates paper pdivisible groups 6. Proceedings of a conference on local fields, springer, 1967. Let r be a complete noetherian local ring, m its maximal ideal. What is a proof that apa is divisible by p where p is.
Some papers and preprints of chingli chai the more recent ones are preprints. Cofund 2012 indam fellowships in mathematics andor. We determine the image of these period morphisms, thereby contributing to a question of grothendieck. Canonical coordinates on leaves of pdivisible groups. Moduli of pdivisible groups international press of boston. Further, every abelian group can be embedded in a divisible group as an essential subgroup in a unique way. The most exciting thing discused will be the relation between connected pdivisible groups and divisible formal lie groups over r. The serretate theorem provides a converse in the case of in. Along the way we obtain a new result on the description of the cohomology of some compact shimura varieties. This construction is based on the moduli theory of pdivisible groups of a fixed isogeny type. 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. Loop spaces, pdivisible groups, and character theory university of notre dame. 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. Patrick walls dieudonn e modules and p divisible groups.
Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups. Galois representations arising from some compact shimura. Therefore all the pdivisible groups on a base scheme s form a category denoted. Statement of purpose the aim here is simply to provide some details to some of the proofs in tates paper t. The period matrices and theta functions of riemann an article which gives a. Are there anypdivisiblegroups over z other than products ofpowers ofgmp and ofqplzp1 2. In addition, nonarchimedean uniformization theorems for. 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. The following main theorem gives the comparison of our theory and the crystalline dieudonn e theory of grothendieck and messing. On period spaces for pdivisible groups sciencedirect. It is fully faithful, if the ideal of nilpotent elements in ris a nilpotent ideal. We prove several results about pdivisible groups and 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. Nontrivial divisible groups are not finitely generated.
There is a very beautiful combinatorial argument that proves the above proposition. Remark there is also an abstract notion of p p divisible group in terms of group schemes. In this monograph padic period domains are associated to arbitrary reductive groups. Associate professor kartik prasanna, cochair professor brian d. Consider all the strings of length mathpmath, where each position of the string can be occupied by one of mathamath distinct alphabets. Suppose a is an abelian variety over perfect eld k of dimension g.
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