Hondatate theory classification of abelian varieties in char p up to isogeny. In particular, two kisogenous abelian varieties, and especially two kdual abelian varieties, either both have, or both have not, good reduction at v. 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. Now back in print, the revised edition of this popular study gives a systematic account of the basic results about abelian varieties. Singularities of divisors on abelian varieties olivier debarre march 20, 2006 this is joint work with christopher hacon. Then in sections 3 and 4 we discuss how each of these three approaches leads to di.
Macular degeneration is one of the visual diseases that can reduce your quality of life. Let us x the choice of a separable closure ks of k. I include punctuation errors which were introduced, but i do not attempt to list all of the original punctuation errors misplaced commas, etc. Furthermore, i know that an embedding in the projective space is equivalent to a riemann form in the complex analytic case. 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. To a simple algebra bwith involution of types iiv and a xed complex representation of b, one attaches the moduli of polarized abelian varieties with endomorphisms of the given type. The page numbering and line numbering below refers to the new version of the book, not. Djview4, an improved djvu viewer for unix, windows, mac. Degeneration of abelian varieties by gerd faltings, 9783540520153, available at book depository with free delivery worldwide. 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. Ben moonen group schemes with additional structures and weyl group cosets.
Degeneration of abelian varieties gerd faltings, ching. 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. Serretate theoremtheory on deformations of pdivisible groups and. Associated to an abelian variety ak is a galois representation which describes the action of the absolute galois group of k on the torsion points of a. Yuguang zhang, balanced embedding of degenerating abelian.
Ben moonen a dimension formula for ekedahloort strata. Endomorphisms of abelian varieties over finite fields john tate cambridge, usa w 1. Indeed, f maps t,a onto a subgroup of finite index of t,a and, if ivj acts trivially on the former, it does also on the latter. Ramanujam and on the mordellweil theorem by yuri manin.
On the equations defining abelian varieties i, ii, iii, inventiones mathematicae, 1966, 1, pp. Folge a series of modern surveys in mathematics on. An abelian variety is simple if it is not isogenous to a product of abelian varieties of lower dimension. Assume that we are given a 1parameter family of principally polarized abelian varieties with theta divisors. Endomorphisms of complex abelian varieties, milan, february 2014. Syzygies of abelian varieties article in journal of the american mathematical society 3 july 2000 with 8 reads how we measure reads. On mumfords construction of degenerating abelian varieties. This is a retinal disease that happens when the macula deteriorates.
Complex abelian varieties christina birkenhake springer. The hard part, the rigidity lemma, is proved for complete algebraic varieties on page 43 of mumfords abelian varieties. In this article, we want to share 7 foods that prevent macular degeneration and are beneficial for your eyes. The decomposition up to isogeny into a product of simple abelian varieties is unique up to isogeny. Abelian varieties with adic galois representation of. By this we will mean that we are in one of the following situations. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics. Onedimensional case let tbe an algebraic torus of dimension one over a eld k. 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. Og be the augmentation ideal defining the identity section of g over s. 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. Typos in the new printing of mumfords \abelian varieties below is a list of typographical errors which i found. The name abelian variety comes from the fact that niels henrik abel constructed such varieties in order to determine values of abelian integrals on riemann surfaces.
It is the analogue in algebraic geometry of the concept of a compact complex lie group. The simple objects of this category are exactly the simple abelian varieties. For those who like quick solutions without having to install software, a variety of free. But for more specialized families we can obtain more. One would like to classify such objects, which, however, is a hopeless task in this generality.
Finally, in x7, we turn to the conditions for tori to be realizable as the degeneration of absolutely simple abelian varieties with nontrivial endomorphisms. Associated to an abelian variety ak is a galois representation which describes the action of the absolute galois group. In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i. Computing isogenies between abelian varieties may be seen as different kind of computational problems depending on the expected input and output of the algorithm. Jun 14, 2016 relaxing spa music, music for stress relief, relaxing music, meditation music, soft music. 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.
Studying a variety over finite fields helps us understand abelian varieties over global. These compactifications have applications to diophantine problems and, of course, are. A selfcontained proof of the existence of the dual abelian variety is given. Milne august 6, 2012 abstract this the original tex. Relaxing spa music, music for stress relief, relaxing music, meditation music, soft music. These are appendices on tates theorem on endomorphisms of abelian varieties over finite fields by c. 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.
Abelian varieties, theta functions and the fourier transform alexander polishchuk. An algebraic construction of an abelian variety with a. If d is not ample, there exists a quotient abelian variety a b such that d is the pullback of a divisor on b. 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. Abelian varieties with maximal galois action on their torsion. The structure of the ring of endomorphisms of an abelian variety is discussed. A polarisation of an abelian variety is an isogeny from an abelian variety to its dual that is symmetric with respect to doubleduality for abelian varieties and for which the pullback of the poincare bundle along the associated graph morphism is ample so it is analogous to a positivedefinite quadratic form. This viewpoint of degeneration of elliptic curves into nice group varieties was also very fruitful in higher dimension as is shown by the work of grothendieck. In the main it follows mumfords book 16 except that most results. A new compactification of the siegel space and degeneration of abelian varieties. A new and complete treatment of semi abelian degenerations of abelian varieties, and their application to the construction of arithmetic compactifications of siegel moduli space, with most of the results being published for the first time. Basics on group schemes, abelian varieties, pdivisible groups.
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. Xy of abelian varieties over kfactors as the composition of a homomorphism and a translation. Pdf on mumfords construction of degenerating abelian. Then jx is isogenous to the product of jy with the prym variety of xy.
Any abelian variety is isogenous to a product of simple abelian varieties. Thus, for instance, by an ascheme we mean a scheme over speca. 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. As an auxiliary tool, mumfords construction uses relatively complete models. Zarhin submitted on 26 jan 1999 v1, last revised apr 1999 this version, v2.
An overview of abelian varieties in homotopy theory. 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. As in the case of abelian varieties, the group scheme structure on a is determined by the zero section mumford 1965, corollary 6. In this appendix he gives a theorem which describes the local structure of a degeneration of abelian surfaces in the spirit of kodairas classification. Abelian varieties are special examples of projective varieties. This book deals with the theory of general abelian varieties and also that of albanese and picard varieties of given varieties. With some more work, we can get classification up to isomorphism. I know that there is an analogy roughly with a prequantum line bundle, but i think the concept of polarization of abelian varieties came first. I a proof for the weil conjecture for abelian varieties. The table of contents has been restored, some corrections have been made,1 there are. As such they can be described by a set of homogeneous polynomial equations.
We prove that a 1dimnl family of abelian varieties with an ample sheaf defining principal polarization can be canonically compactified after a finite base change to a projective family with an ample sheaf. Yellow brick cinema relaxing music recommended for you. Djvu pronounced deja vu is a digital document format with advanced. The theory of abelian varieties originated in the beginning of the ninetheenth centrury with the work of abel and jacobi. Highlights of the book include a classification of semiabelian schemes, construction of the toroidal and the minimal compactification over the integers, heights for abelian. Weil during 19541955 together with the authors own contribution. By introducing a level structure we can trivialize the rational. Pages in category abelian varieties the following 40 pages are in this category, out of 40 total. This is a slightly yellow membrane thats found behind your eyes. Hodge classes on abelian varieties of low dimension authors.
I a proof for the surjectivity in the hondatate theory joint work chingli chai fo. I let a be an abelian surface with multiplication by an inde nite quaternion division algebra over q. Complex abelian varieties the main references here are to the book 67, we brie. We will also classify divisible groups up to isogeny dieudonn. A nonempty subset of the following topics will be covered. 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. Djvu is a webcentric format for distributing documents and images. Dieudonne theory classification of pdivisible groups in char p up to isogeny. Let k be a field, k its algebraic closure, and a an abelian variety.
Abelian varieties with maximal galois action on their. Degenerations of abelian varieties are given by maps g s with s an irre ducible scheme and g a group variety whose generic fibre is an abelian variety. As in the case of abelian varieties, the group scheme structure on a is determined by the zero section 20, corollary 6. Uniruled degenerations of abelian varieties mathoverflow. Many properties of abelian varieties such as the mordellweil theorem extend to pseudoabelian varieties. An overview of abelian varieties in homotopy theory 3 one of the applications in mind has been the construction of. Degeneration of abelian varieties gerd faltings springer. From this formalism, and general facts about abelian varieties, we deduce two results. Djvu is a webcentric format and software platform for distributing documents and images. Endomorphisms of abelian varieties over finite fields. Moduli of abelian varieties texel island, 1999, 255298, progress in math. Degeneration of abelian varieties by gerd faltings, 9783642080883, available at book depository with free delivery worldwide. One can find matsusakas early work on polarizations in his polarized varieties, fields of moduli, and generalized kummer varieties of polarized abelian varieties in american j.
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