Pdf elliptic curves on abelian varieties semantic scholar. In the second half of the course, we shall discuss derived categories and the fouriermukai transform, and give some geometric applications. Newton polygons of abelian varieties over finite fields. Thesearethenotesformath731,taughtattheuniversityofmichigan infall1991,somewhatrevisedfromthosehandedoutduringthecourse.
On mumfords families of abelian varieties request pdf. These are appendices on tates theorem on endomorphisms of abelian varieties over finite fields by c. Typos in the new printing of mumford s \ abelian varieties below is a list of typographical errors which i found. This simpli es many things compared to general varieties, but it also means that one can ask harder questions. 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. The decomposition up to isogeny into a product of simple abelian varieties is unique up to isogeny. Ramanujam and on the mordellweil theorem by yuri manin. In this course, we will cover abelian varieties and divisible groups, also known as barsottitate groups. The most prominent example of such a situation is the case of the group g. Abelian varieties, second edition david mumford download bok. Introduction to abelian varieties and the mumfordtate conjecture. The final prices may differ from the prices shown due to specifics of vat rules. Mumford found abelian surfaces in p4 as zero sets of sections of the horrocksmumford bundle. Anna cadoret and ben moonen integral and adelic aspects of the mumford tate conjecture.
Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. I should mention two other books by the same author, which explore related topics. Dieudonn e theory for such group schemes by using dieudonn e theory for abelian schemes. The group of divisors divc is the free abelian group generated by the set ck. Fp, the moduli space of polarized abelian varieties of dimension g in positive. The morphism is actually a homomorphism since it sends 0 to 0.
Some of the other papers concern the lifting of abelian varieties from characteristic p to characteristic 0 and some concern the theory of locally symmetric spaces which are moduli spaces for various families of abelian varieties. Pdf introduction to abelian varieties download ebook for free. X y be a kmorphism of abelian varieties over k and define h tiy. An introduction to abelian varieties university of utah. For g 1 the two coincide, but for g 2 the right side is much bigger. From this formalism, and general facts about abelian varieties, we deduce two results. The simple objects of this category are exactly the simple abelian varieties. Mumford, an analytic construction of degenerating curves over complete local rings compos. The group law of an abelian variety is necessarily commutative and the variety is nonsingular. For a onedimensional family of abelian varieties equipped with principal. In general, it is not possible to write down explicit 1the case gd2is something of an exception to this statement.
Mumford mentions in a footnote that complete algebraic varieties are automatically irreducible, and he later seems to assume without much explanation that they are connected. We will denote the mumford tate group of awith g mta. Topics include general theorems on abelian varieties, the theorem of the square, divisor classes on an abelian variety, functorial formulas, the picard variety of an arbitrary variety, the iadic representations, and algebraic systems of abelian varieties. As it is a homomorphism between locally free ox modules of the. On mumford s families of abelian varieties on mumford s families of abelian varieties noot, rutger 20010308 00. We first build some basic knowledge and apply it to some interesting problems in number theory. Similarly, if an abelian variety is the solution set of polynomials with. Using expzn, we identify z1nz1 with the ground nc of nth roots of unity in c. This notion, called mukai regularity, is based on mukais concept of fourier transform, and in a very particular form. Typos in the new printing of mumfords \abelian varieties below is a list of typographical errors which i found. In fact mumford tate groups seem to be most useful by far in the case of hodge structures of type 1. Z q k and that k is the total ring of quotients of r. The mumfordtate conjecture for products of abelian varieties.
Complex abelian varieties and theta functions george r. X is a rank r free commutative group, and x xs is a constant group scheme, the. Pdf on mumfords construction of degenerating abelian. These notes are an introduction to the theory of abelian varieties, including the. Every abelian variety of dimension 2is the jacobian variety of a curve of genus 2, and every curve of genus 2has an. It follows easily mumford 1974, corollaries 1 and 2 of section 19 that the algebra of. The completeness condition implies severe restrictions on an abelian variety. We notably determine, for each fibre of such a family over a. The page numbering and line numbering below refers to the new version of the book, not.
The topic is the abelian chabautycoleman and nonabelian chabautykim, balakrishnandogra methods for rigorously finding all rational points on suitable curves over number fields. Thus a major goal of this lecture is to introduce abelian varieties from the hodgetheoretic point of view. Our main reference is abelian varieties, by mumford. Recall that at is the connected component of the identity in the picard variety pic ak of a. Lk is a finite separable field ex tension, a is an abelian variety over l, and a, is the abelian variety over k obtained from a by restriction of scalars. On a principally polarized abelian variety we show how to characterize numerical classes that come from abelian subvarieties of codimension 1 in terms of intersection numbers with the class of the theta divisor. Introduction the easiest way to understand abelian varieties is as higherdimensional analogues of elliptic curves.
We notably determine, for each fibre of such a family over a number field, the possible isogeny types and the possible newton polygons of its reductions. Mumfordtate groups and abelian varieties introduction. Moreover, we define certain homogenous forms on the n\\eronseveri group of the abelian variety that also characterize these classes. Pdf mathematics version of record find, read and cite all the research you need on researchgate. Definition 25 let, be two abelian varieties over of the same dimension. In this way, the 1dimensional case is quite misleading. Then discuss the mumford tate groups, abelian varieties of cmtype and del82, example 3. Now back in print, the revised edition of this popular study gives a systematic account of the basic results about abelian varieties. It includes a discussion of the theorems of honda and tate concerning abelian varieties over finite fields and the paper of faltings in which he proves mordells conjecture. Some remarks on ample line bundles on abelian varieties. Notes on abelian schemes university of south carolina.
The goal of the rst half of this class is to introduce and study the basic structure theory of abelian varieties, as covered in say mumford s book. An elliptic curve is an abelian variety of dimension 1. This is the home page for a reading course to be held at uc san diego during the fall 2019 and winter 2020 quarters, in preparation for the 2020 arizona winter school. An order in k is a subring r of k whose underlying abelian group is isomorphic to zn where n degf. Mumford omits this explanation in 6, so let us explain it more generally. On the equations defining abelian varieties i, ii, iii, inventiones mathematicae, 1966, 1, pp. David mumford was awarded the 2007 ams steele prize for mathematical exposition. Topics in algebraic geometry i abelian varieties bhargav bhatt course description. If you dont want to wait have a look at our ebook offers and start reading immediately.
A complex abelian variety is a smooth projective variety which happens to be a complex torus. 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. Abelian varieties appear naturally as jacobian varieties the connected components of zero in picard varieties and albanese varieties of other algebraic varieties. This induces an action of gon the jacobian jxof xand thus a decomposition of gof jx up to isogeny. Abelian varieties are a natural generalization of elliptic curves to higher dimensions, whose geometry and classification are as rich in elegant results as in the onedimensional ease. A divisorial correspondence induces a morphism of abelian varieties in general, a line bundle on gives a morphism. In this seminar, we will give a rst introduction to abelian varieties. Recovering sheaves from a sheaf on a base 90 part ii. Abelian varieties, second edition david mumford download. Download fulltext pdf on mumford s construction of degenerating abelian varieties article pdf available in tohoku mathematical journal 5 september 1996 with 30 reads. In some cases, a shimura variety may parametrize a family of abelian varieties and in such a case, this circumstance plays an important role in the study of its properties, notably for the construction of canonical models. The hard part, the rigidity lemma, is proved for complete algebraic varieties on page 43 of mumfords abelian varieties. Buy abelian varieties tata institute of fundamental research on free shipping on qualified orders abelian varieties tata institute of fundamental research.
In mathematics, a siegel modular variety or siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. On mumfords families of abelian varieties, journal of pure. Sheaves of abelian groups, and oxmodules, form abelian categories 84 2. The use of theta functions, particularly since mumford s work, has been an important tool in the study of abelian varieties and invertible sheaves on them. Pdf on mumfords construction of degenerating abelian varieties. There is a concise treatment of complex abelian varieties in chapter i of mumford 1970. The topic is the abelian chabautycoleman and nonabelian chabautykim, balakrishnandogra methods for rigorously finding all. Mumford showed that a principally polarized abelian variety can be written as an intersection of explicit quadrics in a. The study of abelian varieties is certainly one of the oldest and richest branches of. The text concludes with a helpful appendix covering the composition of correspondences. Your print orders will be fulfilled, even in these challenging times. Mumford 5 proves this in the case where the eld is algebraically closed. Introduce hodge structures and mention the relation between complex abelian varieties and polarizable hodge structures of type 1.
An sgroup or group scheme over s is a group object in the category of sschemes. Ramanujam and on the mordell weil theorem by yuri manin. In other words, it is an sscheme g equipped with an smap m. This is the only complete book about abelian varieties written from a modern point of view. An abelian variety x over f is called simple over f if any abelian subvariety of xde ned over fis either f0gor x. 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. Abelian fourfold of mumfordtype and kugasatake varieties. Abelian varieties tata institute of fundamental research. Numerous and frequentlyupdated resource results are available from this search. Mumfords elegant yet terse writing style is hard to improve upon. A line bundle on is called a divisorial correspondence if and.
An abelian variety is an algebraic group that is a complete algebraic variety. Pdf introduction to abelian varieties download ebook for. I the harvard community has made this article openly available. More precisely, siegel modular varieties are the moduli spaces of principally polarized abelian varieties of a fixed dimension. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Download fulltext pdf on mumfords construction of degenerating abelian varieties article pdf available in tohoku mathematical journal 5 september 1996 with 30 reads. Ben moonen on the tate and mumford tate conjectures in codimension one for varieties with h 2,0 1. Mumford s elegant yet terse writing style is hard to improve upon. They are named after carl ludwig siegel, a 20thcentury german mathematician who specialized in number theory. I include punctuation errors which were introduced, but i do not attempt to list all of the original punctuation errors misplaced commas, etc. The structure of the ring of endomorphisms of an abelian variety is discussed. Ramanujam, oxford university press india, based on lectures at the tata institute in 196768 1st edition 1970. Moduli of abelian varieties and pdivisible groups chingli chai and frans oort abstract. Introduction to abelian varieties and the mumfordtate.
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