Degeneration of abelian varieties gerd faltings, ching. Get your kindle here, or download a free kindle reading app. Relaxing spa music, music for stress relief, relaxing music, meditation music, soft music. Endomorphisms of abelian varieties over finite fields. Now back in print, the revised edition of this popular study gives a systematic account of the basic results about abelian varieties. The simple objects of this category are exactly the simple abelian varieties. Zarhin submitted on 26 jan 1999 v1, last revised apr 1999 this version, v2. The morphism is actually a homomorphism since it sends 0 to 0. Enter your mobile number or email address below and well send you a link to download the free kindle app. Motivic zeta functions for degenerations of abelian varieties. In fact, we show that the conjecture holds for abelian varieties satisfying some degeneration condition, which generalizes the result for totally degenerate abelian varieties due to gubler and the. Question feed subscribe to rss question feed to subscribe to this rss. The main theorem almost all of the general facts about abelian varieties which we use without comment or refer to as well known are due to weil, and the references for them are 12 and 3.
An analytic construction of degenerating abelian varieties over complete rings, composito math. Sustained pdivisible groups and a foliation on moduli spaces of abelian varieties. Abelian varieties bryden cais a canonical reference for the subject is mumfords book 6, but mumford generally works over an algebraically closed. Let k be a field, k its algebraic closure, and a an abelian variety. This book deals with the theory of general abelian varieties and also that of albanese and picard varieties of given varieties. An introduction to abelian varieties university of utah. In this appendix he gives a theorem which describes the local structure of a degeneration of abelian surfaces in the spirit of kodairas classification. Remarkably, any such thing is an abelian algebraic group. On the number of points on abelian and jacobian varieties over nite elds yves aubry, sa a haloui, gilles lachaud to cite this version. Degenerating abelian varieties 657 in order to justify assertion a of 1. The topic of this book is the theory of degenerations of abelian varieties and its application to the construction of compactifications of moduli spaces of abelian varieties. Xy is a finite etale separable isogeny between two abelian varieties over k. On the number of points on abelian and jacobian varieties over nite elds. Information from the derived categories of coherent sheaves.
Abelian varieties with maximal galois action on their. Abelian varieties tata institute of fundamental research. In this paper we study hodge classes on complex abelian varieties. Weils theorem saying that a rational map from a smooth kscheme to a kgroup scheme of finite type is. Zarhin department of mathematics, pennsylvania state university, university park, ap 16802, usa institute for mathematical problems in biology, russian ademyac of sciences, pushchino, moscow gion,re russia email. Then you can start reading kindle books on your smartphone, tablet, or computer no kindle device required. A new compactification of the siegel space and degeneration of abelian varieties. Degeneration of abelian varieties gerd faltings, chingli chai this book presents a complete treatment of semiabelian degenerations of abelian varieties, and their application to the construction of arithmetic compactifications of siegel moduli space. In this seminar, we will give a rst introduction to abelian varieties. In fact, we show that the conjecture holds for abelian varieties satisfying some degeneration condition, which generalizes the result for totally degenerate.
Aspects of the general theory are then exposed, in particular, local deformation theory of abelian varieties in positive characteristic. Abelian varieties tata institute of fundamental research by david mumford. Lectures on hilbert modular varieties and modular forms. Thus, an abelian variety can be imbedded as a closed subvariety in a projective space.
In other words, ais a smooth projective variety equipped with morphisms. Acta arithmetica, instytut matematyczny pan, 20, 160 3. In this paper some general results on group actions on abelian varieties are given and applied to deduce a decomposition of the jacobian jx for arbitrary group actions. A rank 2 vector bundle on p 4 with 15,000 symmetries with g. These compactifications have applications to diophantine problems and, of course, are also interesting in their own right. The topic of this book is the theory of degenerations of abelian varieties and its. The theory of moduli spaces of abelian varieties with real multiplication is presented first very explicitly over the complex numbers. Mumford describes the analytic methods and results applicable when the ground field k is the complex field c and discusses the schemetheoretic methods and results used to deal with inseparable isogenies when the ground field k has characteristic p. Yellow brick cinema relaxing music recommended for you. Abelian varieties and jacobians november 25, 20 1 abelian varieties and jacobians an abelian variety aover a eld kis an irreducible smooth projective group variety. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. Abelian varieties and weil representations 3 as the automorphism group of a rational padic tate module, which is a vector space over qp rather than something like fpt. Homomorphisms of abelian varieties over nite elds uriy g.
Degeneration of abelian varieties faltings, gerd, chai. Pages in category abelian varieties the following 40 pages are in this category, out of 40 total. Degeneration of abelian varieties by gerd faltings, 9783642080883, available at book depository with free delivery worldwide. These notes are an introduction to the theory of abelian varieties, including the. The completeness condition implies severe restrictions on an abelian variety. An abelian variety can be defined by equations having coefficients in any field. It is used to parametrize the orbits in a finite abelian group under its full automorphism group by a finite distributive lattice. In his book abelian varieties, david mumford defines an abelian variety over an algebraically closed field k k to be a complete algebraic group over k k. A notion of degeneration of elements in groups is introduced. The following proposition shows that, to prove the conjecture for given d and g, it is enough to exhibity one polarized abelian variety, or a suitable degeneration, for. Degeneration of abelian varieties gerd faltings, chingli. Yuguang zhang, balanced embedding of degenerating abelian.
I think this is a surprising theory because it leads to very explicit equations both for individual abelian varieties and for their moduli space. By this we will mean that we are in one of the following situations. This is a survey on motivic zeta functions associated to abelian varieties and calabiyau varieties over a discretely valued field. A nonarchimedean analogue of the calabiyau theorem for totally degenerate abelian varieties liu, yifeng, journal of differential geometry, 2011. We rst show that every algebraic torus over any eld, not necessarily split, can be realized as the special ber of a semiabelian scheme whose generic ber is an absolutely simple abelian variety. Degeneration of abelian varieties gerd faltings springer. Derived equivalence of families of dual abelian varieties.
If you allow semistable reduction and run minimal model program then we have the following result due to fujino which can be applied to semistable degenerations of abelian varieties, calabiyau varieties. Abelian varieties spring quarter, 2015 note to the reader this document consists of lecture notes that tony feng livetexed from a course given by brian conrad at stanford university in the spring quarter of 2015, which both. Degeneration of abelian varieties gerd faltings, chingli chai this book presents a complete treatment of semi abelian degenerations of abelian varieties, and their application to the construction of arithmetic compactifications of siegel moduli space. Zarhin submitted on 26 jan 1999, last revised apr 1999 this version, v2 abstract. Uniruled degenerations of abelian varieties mathoverflow. Abelian varieties, theta functions and the fourier transform thisbookisamodernintroductiontothetheoryofabelianvarietiesandtheta functions.
Another canonical compactification of the moduli space of abelian varieties nakamura, iku, 2010. On the number of points on abelian and jacobian varieties. Advances in knowledge discovery and data mining, part ii. In this expository paper, we survey the various approaches to compactifying moduli stacks of polarized abelian varieties. Endomorphisms of abelian varieties over finite fields john tate cambridge, usa w 1. Geometric bogomolov conjecture for abelian varieties and some. Algebraic tori as degenerations of abelian varieties 3 in case t has dimension one, we can exhibit the desired semi abelian schemes gover dedekind sfor each nonsplit torus texplicitly. Jun 14, 2016 relaxing spa music, music for stress relief, relaxing music, meditation music, soft music.
Abelian varieties of dimension 1 see also 14kxx 14c25. A divisorial correspondence induces a morphism of abelian varieties in general, a line bundle on gives a morphism. Then f is a galois cover with covering group kerfk. Finding ebooks booklid booklid download ebooks for free. Sections of this bundle define an abelian surface in p 4. Yigeng zhao wednesday, 1012 h, m 006 introduction the aim of this seminar is to study some basic theories of abelian varieties over an algebraic closed eld k, which are one of the most important and most studied objects in arithmetic geometry. In mathematics, particularly in algebraic geometry, complex analysis and algebraic number.
The decomposition up to isogeny into a product of simple abelian varieties is unique up to isogeny. Projective regular models for abelian varieties, semistable reduction, and the height pairing. The assumption of connectedness is necessary for that conclusion. The content of the book is absolutely amazing actually among the best writings i have ever read, its like a more detailed version of delignes writing, and in english. Hodge classes on abelian varieties of low dimension authors. Chai, chingli 1990, degeneration of abelian varieties, springer verlag. Motivic zeta functions for degenerations of abelian. An introduction to abelian varieties stefano filipazzi august 31, 2016 these notes are supposed to be a handout for the student seminar in algebraic geometry at the university of utah. Definition 25 let, be two abelian varieties over of the same dimension. From this formalism, and general facts about abelian varieties, we deduce two results. A line bundle on is called a divisorial correspondence if and. Abelian varieties, theta functions and the fourier transform.
Expanded version of a plenary talk at the iccm annual meeting in taipei. We explain how they are related to denef and loesers motivic zeta function associated to a complex hypersurface singularity and we investigate the relation between the poles of the zeta function and the eigenvalues of the monodromy action on the. Is there a notion of hyperbolicity for number rings. Algebraic tori as degenerations of abelian varieties kaiwen lan and junecue suh abstract.
Hodge classes on abelian varieties of low dimension. Abelian varieties with maximal galois action on their torsion points david zywina. A pictorial description of this lattice leads to an intuitive selfcontained exposition of some of the basic facts concerning these orbits, including. Cm liftings of abelian varieties frans oort arithmetic algebraic geometry a conference on the occasion of thomas zinks 60th birthday talk bielefeld vi2009 in this talk we study cm liftings of abelian varieties from a. The interaction of an ample line bundle with the group structure on an abelian variety is the subject of the first paper below as well as volume three of my tata lectures on theta. Model program then we have the following result due to fujino which can be applied to semistable degenerations of abelian varieties, calabiyau varieties. Weils theorem saying that a rational map from a smooth kscheme to a kgroup scheme of finite type is a morphism if it is defined in codimension 1. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext.