Mumford let qalg be the algebraic closure of q in c, let sbe an irreducible algebraic curve over qalg, and let gs be a semiabelian scheme over s of relative dimension 2 and toric rank 1. All my papers in this field have been published by springerverlag in two volumes, a selected papers on the classification of varieties and moduli spaces, and b selected papers ii, on algebraic geometry including correspondence with grothendieck. An sgroup or group scheme over s is a group object in the category of sschemes. Pdf introduction to abelian varieties download ebook for free. The definition is based on the fouriermukai transform, and in a special case depending on the choice of a polarization it parallels and strenghtens the usual castelnuovo mumford regularity. The group law of an abelian variety is necessarily commutative and the variety is nonsingular. It gives a systematic account of the basic results about abelian varieties. A in the case where the abelian variety a is isogenous to a product of simple abelian varieties of type gsp, that means that they are simple abelian varieties such that the mumfordtate group is isogenous to the group of symplectic similitudes, and such that the mumford. For abelian varieties such as a p, there is a definition of local zetafunction available. Mumford let qalg be the algebraic closure of q in c, let sbe an irreducible algebraic curve over qalg, and let gs be a semi abelian scheme over s of relative dimension 2 and toric rank 1. G such that the usual group axiom diagrams commute.
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. Complex abelian varieties and theta functions george r. Ben moonen on the tate and mumfordtate conjectures in codimension one for varieties with h 2,0 1. On mumfords families of abelian varieties citeseerx. Thus, for instance, by an ascheme we mean a scheme over speca. Curves and their jacobians download ebook pdf, epub. 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. An elliptic curve is an abelian variety of dimension 1. 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 this way, the 1dimensional case is quite misleading. Download it once and read it on your kindle device, pc, phones or tablets. A divisorial correspondence induces a morphism of abelian varieties in general, a line bundle on gives a morphism.
Get your kindle here, or download a free kindle reading app. The study of abelian varieties is certainly one of the oldest and richest branches of. On mumfords families of abelian varieties request pdf. 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. Download 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. Number theory currently has at least three different perspectives on nonabelian phenomena. Abelian fourfold of mumfordtype and kugasatake varieties.
Pdf introduction to abelian varieties download ebook for. These are appendices on tates theorem on endomorphisms of abelian varieties over finite fields by c. Cubic threefolds and abelian varieties of dimension five by sebastian casalainamartin. For g 1 the two coincide, but for g 2 the right side is much bigger. 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.
Mumfordtate groups and abelian varieties 3 c the hodge classes in vm are the twisted mtvinvariants. This is a reprinting of the revised second edition 1974 of david mumford s classic 1970 book. Use features like bookmarks, note taking and highlighting while reading abelian varieties. Ben moonen on the tate and mumford tate conjectures in codimension one for varieties with h 2,0 1. In other words, it is an sscheme g equipped with an smap m. A line bundle on is called a divisorial correspondence if and. On the mordell weil rank 263 to prove this, write the extension as l, c l2, with l3l, unramified of degree f and l,l, totally ramified of degree e. We will denote the mumfordtate group of awith g mta. Thus a major goal of this lecture is to introduce abelian varieties from the hodgetheoretic point of view. 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. In 2010, the same authors gave an explicit formula of the invariant. It includes expositions of analytic methods applicable over the ground field of complex numbers, as well as of schemetheoretic methods used to deal with inseparable. Pdf on mumfords construction of degenerating abelian. We introduce the notion of mukai regularity mregularity for coherent sheaves on abelian varieties.
From this formalism, and general facts about abelian varieties, we deduce two results. Anna cadoret and ben moonen integral and adelic aspects of the mumfordtate conjecture. The hard part, the rigidity lemma, is proved for complete algebraic varieties on page 43 of mumfords abelian varieties. The gottingen state and university library provides access. On the mumfordtate conjecture for abelian varieties s. Mumford defined a class of shimura varieties all of which parametrize a family of abelian varieties. On the mumfordtate conjecture for abelian varieties. This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their jacobians, and of abelian varieties all of them defined over a complete nonarchimedean. Definition 25 let, be two abelian varieties over of the same dimension. Introduction to abelian varieties and the mumfordtate conjecture. In this seminar, we will give a rst introduction to abelian varieties.
David mumford was awarded the 2007 ams steele prize for mathematical exposition. Introduction the easiest way to understand abelian varieties is as higherdimensional analogues of elliptic curves. The mumfordtate conjecture is a precise way of saying that certain extra. The mumfordtate conjecture for products of abelian varieties. Most of these can be posed for an abelian variety a. Introduction to abelian varieties and the mumfordtate. Abelian varieties appear naturally as jacobian varieties the connected components of zero in picard varieties and albanese varieties of other algebraic varieties.
Curves and their jacobians download ebook pdf, epub, tuebl. The mumfordtate conjecture is a precise way of saying that. Pdf on mumfords construction of degenerating abelian varieties. Moreover, varieties of mumfordtype gave the first example of abelian varieties not char. Cubic threefolds and abelian varieties of dimension five. On the equations defining abelian varieties i, ii, iii, inventiones mathematicae, 1966, 1, pp. The purpose of this talk is to collect together what seem to me to be the most basic moduli spaces for curves and abelian varieties. 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. Now back in print, the revised edition of this popular study gives a systematic account of the basic results about abelian varieties. Pdf mathematics version of record find, read and cite all the research you need on researchgate. Download fulltext pdf on mumfords construction of degenerating abelian varieties article pdf available in tohoku mathematical journal 5 september 1996 with 30 reads.
Somewhat oversimplifying, one can say that these families are characterized by the hodge classes living on the powers of the abelian varieties. Anna cadoret and ben moonen integral and adelic aspects of the mumford tate conjecture. We will denote the mumford tate group of awith g mta. Non abelian fundamental groups and iwasawa theory download. Abelian varieties, second edition david mumford download. The use of theta functions, particularly since mumfords work, has been an important tool in the study of abelian varieties and invertible sheaves on them. David mumford, on the equations defining abelian varieties i invent. Cubic threefolds and abelian varieties of dimension five by.
The mumford tate conjecture for products of abelian varieties. We write g tor for the union of all the torsion points of the various bers of g. The morphism is actually a homomorphism since it sends 0 to 0. I the harvard community has made this article openly available. Ramanujam and on the mordellweil theorem by yuri manin. Buy abelian varieties tata institute of fundamental research on free shipping on qualified orders abelian varieties tata institute of fundamental research. Similarly, if an abelian variety is the solution set of polynomials with. Abelian varieties tata institute of fundamental research. Mumford, curves and their jacobians ann arbor, mich d. Kempf, complex abelian varieties and theta functions springer which is not bad, though it is not errorfree and the approach taken is not the one i propose to take. The decomposition up to isogeny into a product of simple abelian varieties is unique up to isogeny. Abelian varieties, second edition david mumford download bok.
The definition is based on the fouriermukai transform, and in a special case depending on the choice of a polarization it parallels and strenghtens the usual castelnuovomumford regularity. We write mtv for g1 g2 and call it the mumfordtate group of v. I also used various other more specialised references when, for example, trying to fathom out the hilbert case and so on. The structure of the ring of endomorphisms of an abelian variety is discussed. In mathematics, the concept of abelian variety is the higherdimensional generalization of the. 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. In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. Moduli of abelian varieties and pdivisible groups chingli chai and frans oort abstract. In the main it follows mumford s book 16 except that most results are stated relative to an arbitrary base field, some additional. Torsion for abelian varieties of type iii sciencedirect. Fp, the moduli space of polarized abelian varieties of dimension g in positive. Mumfordtate groups and abelian varieties introduction. Tankeev journal of mathematical sciences volume 81, pages 2719 2737 1996 cite this article. Let ui be the extention of u1 to li for i 2,3 and g the different of ll.
In fact mumford tate groups seem to be most useful by far in the case of hodge structures of type 1. On mumfords families of abelian varieties sciencedirect. Mumford showed that a principally polarized abelian variety can be written as an intersection of explicit. In the second half of 2009, experts from each of these three areas.
Mumford, tata lecture notes on theta i birkh auser another fairly modern book on abelian varieties is g. The simple objects of this category are exactly the simple abelian varieties. My goal was a concrete reference for all the wellknown explicit bijections be. Shimura varieties all of which parametrize a family of abelian varieties. Abelian varieties tata institute of fundamental research by david mumford author, c.
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