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 the mordelllang conjecture barry mazur abstract. Thesearethenotesformath731,taughtattheuniversityofmichigan infall1991,somewhatrevisedfromthosehandedoutduringthecourse. In 12, rodr guez completely describes abelian varieties admitting an involution.
Abelian fourfold of mumfordtype and kugasatake varieties. Abelian varieties tata institute of fundamental research. Some remarks on ample line bundles on abelian varieties. The page numbering and line numbering below refers to the new version of the book, not.
In this course, we will cover abelian varieties and divisible groups, also known as barsottitate groups. Pdf a survey around the hodge, tate and mumfordtate. The structure of the ring of endomorphisms of an abelian variety is discussed. Pdf this survey describe hodge, tate and mumfordtate conjectures for abelian varieties. Abelian varieties with adic galois representation of.
An analytic construction of degenerating abelian varieties over complete rings by david mumford compositio mathematica, vol. On the mumfordtate conjecture for abelian varieties with. Topics in algebraic geometry i abelian varieties bhargav bhatt course description. On hrushovskis proof of the maninmumford conjecture. Some papers and preprints of chingli chai penn math. Pdf on the newton polygons of abelian varieties of mumford. I think this is a surprising theory because it leads to very explicit equations both for individual abelian varieties and for their moduli space. Abelian varieties with ladic galois representation of. This is the only complete book about abelian varieties written from a modern point of view. We show that these conditions force the image of the representations to be big, so that the mumford tate conjecture.
An abelian variety is simple if it is not isogenous to a product of abelian varieties of lower dimension. An abelian motive over k is a summand of a tate twist of the motive of an abelian variety over k. Typos in the new printing of mumfords \abelian varieties below is a list of typographical errors which i found. I include punctuation errors which were introduced, but i do not attempt to list all of the original punctuation errors misplaced commas, etc. The complex mumfordtate group of x is isogenous to sl2 3. On adic compatibility for abelian motives the mumford. Kevin buzzard april 26, 2012 last modi ed august 2005. Abelian varieties talk 2 chris nicholls june 3, 2016 1 theorem of the cube we follow mumfords abelian varieties book. A good reference for today is mumfords abelian varieties mr282985 or milnes notes. Abelian varieties with adic galois representation of mumford s type rutger noot abstract this paper is devoted to the study of 4dimensional abelian varieties over number elds with the property that the lie algebra of the image of some associated adic galois representation is q isomorphic to c sl 23. Milne august 6, 2012 abstract this the original tex.
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. Let xand y be complete varieties, and zany variety, and let x 0. The study of abelian varieties is certainly one of the oldest and richest branches of. Some remarks on ample line bundles on abelian varieties springerlink advertisement. In the second half of the course, we shall discuss derived. 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 mumfords book. To be precise, it is really w in 2 that is the abelian variety, but since w is unique and functorial in xit is permissible to abuse terminology and call xthe abelian variety. For abelian varieties we indeed know some examples where it is true. For abelian varieties of cm type this conjecture follows from theorem a. The hard part, the rigidity lemma, is proved for complete algebraic varieties on page 43 of mumfords abelian varieties. A complex abelian variety is a smooth projective variety which happens to be a complex torus. On mumford s families of abelian varieties rutger noot abstract in mum69, mumford constructs families of abelian varieties which are parametrized by shimura varieties but which are not of pel type. Reading course on abelian varieties in this reading course, the participants learn some basic results about abelian varieties, following mostly d.
The simple objects of this category are exactly the simple abelian varieties. A good reference for today is mumford s abelian varieties. An introduction to abelian varieties university of utah. In this seminar, we will give a rst introduction to abelian varieties. To get an lfunction for a itself, one takes a suitable euler product of such local functions. 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. In the main it follows mumford s book 16 except that most results are stated relative to an arbitrary base field, some additional results are proved, and etale cohomology is included. This is the only complete book about abelian varieties written from a modern. In this thesis we study galois representations corresponding to abelian varieties with certain reduction conditions. Ben moonen on the tate and mumford tate conjectures in codimension one for varieties with h 2,0 1. Mumfordtate groups and abelian varieties introduction.
On mumfords families of abelian varieties request pdf. 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. An excellent reference for the basics of this theory is mumford 1974. The pdf version contains the table of contents as bookmarks, which allows easy navigation. A good reference for today is mumford s abelian varieties mr282985 or milnes notes.
An excellent reference for this material is mumford s book on abelian varieties, see. From this formalism, and general facts about abelian varieties, we deduce two results. If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact web. The pdfversion contains the table of contents as bookmarks, which allows easy navigation. David mumford was awarded the 2007 ams steele prize for mathematical exposition. They play a useful role in the proof of the hecke orbit conjecture for siegel modular varieties. In the same spirit, the present works makes an attempt to. Abelian varieties with adic galois representation of mumford.
In practice this means that an abelian motive m is a package. Let x be a complex abelian fourfold of mumfordtype and let v h 1 x. 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. The assumption of connectedness is necessary for that conclusion. This is an introductory exposition to background material useful to appreciate various formulations of the mordelllang conjecture now established by recent spectacular work due to vojta, faltings, hrushovski, buium, voloch, and others. An abelian variety can be defined by equations having coefficients in any field. In fact mumford tate groups seem to be most useful by far in the case of hodge structures of type 1. Anna cadoret and ben moonen integral and adelic aspects of the mumford tate conjecture.
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. Abelian varieties and moduli donu arapura april 19, 2012. 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. Similarly, if an abelian variety is the solution set of polynomials with. In the second half of the course, we shall discuss derived categories and the fouriermukai transform, and give some geometric applications. Ramanujam and on the mordellweil theorem by yuri manin.
Abelian varieties with adic galois representation of mumfords type rutger noot abstract this paper is devoted to the study of 4dimensional abelian varieties over number elds with the property that the lie algebra of the image of some associated adic galois representation is q isomorphic to c sl 23. We first build some basic knowledge and apply it to some interesting problems in number theory. Introduction to abelian varieties and the mumfordtate conjecture. Work on these articles have been supported by the national science foundation since 1990, including the following grants. After some preliminaries on endomorphism ring, polarization. Suppose that lis a line bundle on x y z whose restrictions to each of fx 0g y z, xf y 0g zand x yf z. It seems very natural to generalise the conjecture to arbitrary smooth projective varieties, and i think nowadays most people mean the general version when referring to the conjecture. Our main reference is abelian varieties, by mumford. Pseudoeffective and nef classes on abelian varieties volume 147 issue 6 olivier debarre, lawrence ein, robert lazarsfeld, claire voisin. The mumfordtate conjecture for products of abelian varieties. Let mg be the moduli space of curves of genus g over k.
Originally the mumfordtate conjecture was formulated for abelian varieties. Pseudoeffective and nef classes on abelian varieties. As a preparation for the meeting, the participants are required to read each week a certain portion of the literature, described below. Any abelian variety is isogenous to a product of simple abelian varieties. Such complex tori are called abelian varieties over c. If l and l0are two ample line bundles on an abelian variety ak, then l d l0if and only if l0t x l for some point x 2a. Noot studies the galois representations associated to an abelian variety over a number field which appears as a closed fiber of a mumford s family, and he. For abelian varieties such as a p, there is a definition of local zetafunction available. Mumford, d curves on an algebraic surface to appear in annals of math. 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. Finally, we try to work out speci c group actions on abelian varieties. Other readers will always be interested in your opinion of the books youve read.
The irreducibility of the space of curves of given genus by p. The decomposition up to isogeny into a product of simple abelian varieties is unique up to isogeny. These are appendices on tates theorem on endomorphisms of abelian varieties over finite fields by c. Thus a major goal of this lecture is to introduce abelian varieties from the hodgetheoretic point of view. Remarkably, any such thing is an abelian algebraic group. Canonical compactifications of moduli spaces for abelian varieties. V in order that t v be imbeddable in projective space, or equivalently, that t have the structure of abelian variety over c.
This simpli es many things compared to general varieties, but it also means that one can ask harder questions. These are some notes i wrote in order to teach myself the classical analytic theory of moduli spaces for abelian varieties. The main purpose of this survey is to give a complete account of the computational aspects of the isogenies of low dimensional abelian varieties and their use in cryptography. Abelian varieties and the mordelllang conjecture department of. I the harvard community has made this article openly available. A note on the mumford tate conjecture for cm abelian varieties yu, chiafu, taiwanese journal of mathematics, 2015. Abelian varieties and the mordelllang conjecture 203 i. Jun 17, 2011 a mumfords family, and he obtains various results about such an abelian variety, notably potentially go od reduction, classi. The topic is the abelian chabautycoleman and nonabelian chabautykim, balakrishnandogra methods for rigorously finding all rational points on suitable curves over number fields. 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 varieties, second edition david mumford download. The irreducibility of the space of curves of given genus. As in the case of abelian varieties, the group scheme structure on a is determined by the zero section mumford 1965, corollary 6.
Next i prove the weak mordellweil theorem, as the same ideas will be important for us later on. In the main it follows mumford s book 16 except that most results are stated relative to an arbitrary base field, some additional. Nevertheless we will construct one and only one new compacti. Here the compactification ag constructed by mumford and its properties are explained. They may well contain mistakes, and they might have a \lopsided feel because they emphasize only the parts. Some papers and preprints of chingli chai the more recent ones are preprints. From the modern viewpoint, the theory of abelian functions is an analytic aspect of the theory of abelian varieties, that is, projective algebraic group varieties. An analytic construction of degenerating abelian varieties. Compactification by gitstability of the moduli space of abelian varieties nakamura, iku, 2016. The main result of this note is that mg is irreducible for every k. We recover information about the hodge structure of x using representations of the lie algebras sl 2 3 and sp 8 acting on v. We will denote the mumford tate group of awith g mta. In this note we give another proof of this result and extend it to cm motives.
The mumford tate conjecture is first proved for cm abelian varieties by h. In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. 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. 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. Introduction to abelian varieties and the mumfordtate. 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. In this expository paper, we survey the various approaches to compactifying moduli stacks of polarized abelian varieties. The younger generation scarcely knows what abelian functions are.