The study of equicharacteristic deformations does not give information concerning how much ramification is needed to lift an abelian variety. Mumford tate conjecture for abelian varieties over number. Kuga proved the hodge conjecture for such varieties in k, 2. There is a concise treatment of complex abelian varieties in chapter i of mumford 1970. 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. On mumford s construction of degenerating abelian varieties valery alexeev and iku nakamura received may 8, 1998 abstract. This is the only complete book about abelian varieties written from a modern. Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. The page numbering and line numbering below refers to the new version of the book, not.
This is the only complete book about abelian varieties written from a modern point of view. We refer to mumford 1984 for fundamental facts about hyperelliptic curves and their. Mumford 5 proves this in the case where the eld is algebraically closed. A good reference for today is mumford s abelian varieties mr282985 or milnes notes. 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. This notion, called mukai regularity, is based on mukais concept of fourier transform, and in a very particular form called theta regularity it parallels and strengthens the usual castelnuovo mumford regularity with respect to polarizations on abelian varieties. An introduction to abelian varieties university of utah.
In fact mumford tate groups seem to be most useful by far in the case of hodge structures of type 1. For a onedimensional family of abelian varieties equipped with principal. Complex multiplication of abelian varieties and its applications to number theory. Torsion for abelian varieties of type iii sciencedirect. Line bundles and cohomology of complex tori emre can sert. With an eye or two towards applications to pells equation and to davenports work on integration of algebraic functions, umberto zannier and i have recently charac. Mumfordtate groups and abelian varieties introduction. Pdf a new compactification of the siegel space and.
Ag 24 feb 2009 donu arapura and manish kumar given a smooth not necessarily proper complex variety u, beilinson b conjec tured that all hodge cycles in h. Typos in the new printing of mumford s \ abelian varieties below is a list of typographical errors which i found. Many of the results about abelian varieties over c continue to hold over k. Let xbe a desingularization of y double cover of p3 with equation u2 fx 0x 3, where degf 4 and the quartic f 0 has 10 nodes in special position. Abelian varieties bryden cais a canonical reference for the subject is mumford s book 6, but mumford generally works over an algebraically closed. Some cases of the mumfordtate conjecture and shimura varieties.
Download and read online complex multiplication of abelian varieties and its applications to number theory ebooks in pdf, epub, tuebl mobi, kindle book. As a preparation for the meeting, the participants are required to read each week a certain portion of the literature, described below. In general, it is not possible to write down explicit 1the case gd2is something of an exception to this statement. Mumfordtate groups and hodge classes on abelian varieties of low. X y be a kmorphism of abelian varieties over k and define h tiy fex. Their relation is analogous to that between stable curves and nonsingular curves. General theory of abelian varieties and their moduli.
Such complex tori are called abelian varieties over c. Pdf beilinsonhodge cycles on semiabelian varieties. If is a rational prime distinct from the characteristic of k, the. These notes are an introduction to the theory of abelian varieties, including the. I include punctuation errors which were introduced, but i do not attempt to list all of the original punctuation errors misplaced commas, etc.
Mumford constructs in mum67 the universal abelian variety x g,n a g,n with a totally symmetric normalized relatively ample. In this note, this result is extended to abelian varieties of higher dimensions and some connected. U, q come from motivic cohomology, or more precisely that the so called regulator map reg. Counting points on curves and abelian varieties over. Dec 18, 2000 moreover, varieties of mumford type gave the first example of abelian varieties not char acterized by their endomorphism algebra and having a small mumford tate group, in the sense just explained. The theory of abelian varieties originated in the beginning of the ninetheenth centrury with the work of abel and jacobi. One wants the rami cation curve to be the union of. Mar 08, 2001 then, for each possibility listed in table 1, there exist an extension f of k, a prime p f of f lying over p, with residue field k, and a point x.
Abelian variety a over a finite field of q elements is explicitly given as an irreducible closed set. Several conjectures were formulated, including an analog of the mumford tate conjecture for ordinary abelian varieties. For a onedimensional family of abelian varieties equipped with principal theta divisors a canonical limit is constructed as a pair consisting of a reduced projective variety and a cartier divisor on it. The aim of this paper is to prove a theorem of axlindemann type for complex semi abelian varieties as an application of a big picard theorem proved by the author in 1981, and then apply it to prove a theorem of classical manin mumford conjecture for semi abelian varieties, which was proved by m. It follows easily mumford 1974, corollaries 1 and 2 of section 19 that the algebra of endomorphisms of the abelian variety a, after being tensored with q, end ab var f a zq. Reading course on abelian varieties in this reading course, the participants learn some basic results about abelian varieties, following mostly d. Stable birational invariants and the luroth problem. Abelian varieties, ladic representations, and lindependence. On mumfords families of abelian varieties sciencedirect. Could someone please point me towards a proof of why the image of a galois representation on the tatemodule of an abelian variety is limited by its mumford tate group.
Abelian fourfold of mumfordtype and kugasatake varieties. Pink let a be an abelian variety of dimension g over a global. We prove the mumford tate conjecture for those abelian varieties over number fields whose extensions to c have attached adjoint shimura varieties that are products of simple, adjoint shimura varieties of certain shimura types. Unirational stably irrational varieties artin mumford double solid. These conjectures hold in the case of hilbert modular varieties. Degeneration of abelian varieties gerd faltings springer. The dual abelian variety, polarisations, and the weil pairing 18 7. An abelian variety x over f is called simple over f if any abelian subvariety of xde ned over fis either f0gor x. Mumford omits this explanation in 6, so let us explain it more generally.
An icon used to represent a menu that can be toggled by interacting with this icon. We will be following the more analytic side of the subject using both cornell and lange et. Mumford, david 2008 1970, abelian varieties, tata institute of. Now back in print, the revised edition of this popular study gives a systematic account of the basic results about abelian varieties. A group scheme over a base scheme s is an sscheme g. For abelian varieties such as a p, there is a definition of local zetafunction available. Complex abelian varieties christina birkenhake springer. In mathematics, particularly in algebraic geometry, complex analysis and algebraic number. As in the case of abelian varieties, the group scheme structure on a is determined by the zero section mumford 1965, corollary 6. Recall that the generalized jacobian variety of a stable curve is semiabelian 4 theorem 2. 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. An automorphismf of an abelian varietyx is called fixed point free if it admits no fixed points other than the origin and this is of multiplicity one. Just as elliptic curves have a natural moduli space, over characteristic 0 constructed as a quotient of the upperhalf plane by the action of, there is an analogous construction for abelian varieties using the siegel upper halfspace and the symplectic group. 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.
Similarly, if an abelian variety is the solution set of polynomials with. To get an lfunction for a itself, one takes a suitable euler product of such local functions. Beilinsonhodge cycles on semiabelian varieties arxiv. Complex abelian varieties and theta functions george r. Fixedpoint free automorphisms of 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. X is a rank r free commutative group, and x xs is a constant group scheme. Get free complex multiplication of abelian varieties and its applications to number theory textbook and unlimited access to our library by created an account. We will also use this invariant to locate a cm abelian fourfold that is obtained by iterating the hodge structure of the jacobian of a hyperelliptic curve, which was studied by shioda and was conjectured and disproved by the author to be a special case of mumford s constructions. Reductions of abelian varieties of generalized mumford type. Abelian varieties, ladic representations, and lindependence m. Every abelian variety of dimension 2is the jacobian variety of a curve of genus 2, and every curve of genus 2has an.
This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, their jacobites, and the types of abelian all defined in a highly valuable nonconventional context. Mf such that the corresponding abelian variety x has mumford tate group g and such that x has good reduction x k with the given newton polygon and of the given isogeny type. We will denote the mumford tate group of awith g mta. Notes on abelian schemes university of south carolina. For the basic theory of abelian varieties an excellent reference is 2. Abelian varieties are special examples of projective varieties. Abelian varieties are a natural generalization of elliptic curves to higher. The study of abelian varieties is certainly one of the oldest and richest branches of. In the classical case, it is not di cult to show that topologically an abelian variety is a complex torus. It is well known that the elliptic curve withjinvariant 0 is the only elliptic curve admitting a fixed point free automorphism. An sqav is a compactification of a semiabelian variety i. As such they can be described by a set of homogeneous polynomial equations.
Thus a major goal of this lecture is to introduce abelian varieties from the hodgetheoretic point of view. I the harvard community has made this article openly available. Cubic threefolds and abelian varieties of dimension five. 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. There is a concise treatment of complex abelian varieties in chapter i of mumford. One would like to classify such objects, which, however, is a hopeless task in this generality. The use of theta functions, particularly since mumfords work, has been an. Dieudonn e theory for such group schemes by using dieudonn e theory for abelian schemes. More precisely, the result 1 implies that for a polarized abelian variety x0, x. Introduction to abelian varieties and the mumfordtate. Abelian varieties a canonical reference for the subject is.
Introduction to abelian varieties and the mumfordtate conjecture. Some remarks on ample line bundles on abelian varieties. An abelian variety is a group variety, which as a variety, is complete. At parts i have expanded the explanations to suit my taste, especially in the first two sections.
Complex multiplication of abelian varieties and its. Milnes article 4 is also a good source and allows a general base. For abelian varieties of dimension g in projective n space over fq, we. In this seminar, we will give a rst introduction to abelian varieties.
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