These suggest that the indecomposability of vector bundles over projective space is generally very erratic and hard to predict. Throughout curve means smooth projective curve over an algebraically closed field. We undertake a detailed study of such hecke modi cations. Elliptic curves, vector bundles on curves, topology of real algebraic varieties. From here, we go on to classify indecomposable rank two vector bundles over an elliptic curve. In 1957, sir michael atiyah provided a classi cation of vector bundles over elliptic curves. Here we study vector bundles on elliptic curves over a dvr. The notion is motivated by the deformation theory of curves in projective space incident to given.
Vector bundles on elliptic curves over a discrete valuation. The best known algorithm to solve the ecdlp is exponential, which is why elliptic curve groups are used for cryptography. Rank 3 lagrangian vector bundles on an elliptic curve. The group of exact autoequivalences of the derived category dbcohe. Because of constraints of space and time, we do not give complete proofs of all of the results, and this chapter is mainly intended as a sampler of various methods for studying bundles over x. Vector bundles over an elliptic curve, proceedings of the london mathematical society, volume s37, issue 1, 1 january 1957, pages 414452, s. Moduli spaces of hecke modi cations for rational and elliptic. We shall be concerned with vector bundles over x, i. Ample vector bundles on curves nagoya mathematical journal. Elliptic curves, second edition dale husemoller springer springer new york berlin heidelberg hong kong london milan paris tokyo. We also describe explicit morphisms of vector bundles that represent all possible hecke modi cations of all possible rank 2. Holomorphic principal bundles over elliptic curves ii. Rank two vector bundles on elliptic curves ciara daly abstract.
He has shown that the category of vector bundles on an. This kind of jumping of numerical invariants does not happen in moduli spaces of stable vector bundles. Vector bundles over an elliptic curve atiyah 1957 proceedings. In particular, we classify the vector bundles whose restriction to the special fiber is stable. The problem of classification of vector bundles over an elliptic curve was con sidered and completely solved by atiyah in 1. How do you describe vector bundles on elliptic curves.
Vector bundles over an elliptic curve 415 embedded biregularly in some projective space. Alexandre zotine, simon fraser university global generation of vector bundles over elliptic curves given an algebraic set x e. Coherent state transforms and vector bundles on elliptic curves. The result of grothendieck 1957, that holomorphic vector bundles on the riemann sphere are sums of line bundles, is now often called the birkhoffgrothendieck theorem, since it is implicit in much earlier work of birkhoff 1909 on the riemannhilbert problem. Indecomposable vector bundles on smooth elliptic curves were classi. An excuse for me is that some of his results are reformuleted in a strenghened form using the language of. Vector bundles over a real elliptic curve contents 1. We point out that algebraic varieties of the form x. Vector bundles on the moduli stack of elliptic curves.
In the case of a projective algebraic variety the converse is also true. Interpolation and vector bundles on curves abstract interpolation is a property of vector bundles on curves closely related to slope stability. Motivation and background i read somewhere that atiyah has classified vector bundles on elliptic curves. Vector bundles on elliptic curves mathematics stack exchange. Let xbe a noetherian scheme, and ebe a coherent sheaf over x.
In 1957, sir michael atiyah showed that every indecomposable bundle over a smooth elliptic curve was determined by a point on the curve, and. Vector bundles and torsion free sheaves on degenerations of. If k is the complex field then it has been shown by serre 9 that the algebraic and. Every algebraic subvariety of an algebraic variety over cde. Coherent state transforms and vector bundles on elliptic curves carlos a. Alexandre zotine, simon fraser university global generation. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix problems or via fouriermukai transforms, both methods are discussed here. Vector bundles over weighted projective lines 11 5. Vector bundles on the moduli stack of elliptic curves core. Vector bundles over elliptic surfaces springerlink. The aim of this paper is to give an overview of rank two vector bundles on an elliptic curve. A slope of a holomorphic vector bundle w over a nonsingular algebraic curve or over a riemann surface is a rational number.
These bundles can be explicitly represented via transition matrices, which encode how the vector spaces vary as one moves along the variety. Springer new york berlin heidelberg hong kong london milan paris tokyo. Atiyah 1957 gave the classification of vector bundles on elliptic curves. Vector bundles of degree zero over an elliptic curve, flat bundles. For singular genus one curves over a dvr, we consider the same problem for. Vector bundles and torsion free sheaves on degenerations of elliptic curves 5 theorem 4 see bk1.
This chapter gives a variety of techniques for understanding stable rank 2 vector bundles v over elliptic surfaces x. Let e be an irreducible projective curve of arithmetic genus one over an algebraically closed. Since it is known that onedimensional complex tori correspond to elliptic curves and since the classi. Vector bundles on elliptic curves over a discrete valuation ring. Vector bundles on degenerations of elliptic curves and yang. Vector bundles on singular projective curves springerlink. Ample vector bundles on curves volume 43 robin hartshorne. In works of burban, drozd and greuel it was shown that the categories of vector bundles and coherent sheaves on cycles of projective lines are tame. Full text views reflects the number of pdf downloads, pdfs sent to. The absolutely indecomposable vector bundles on elliptic curves over algebraically closed fields are determined by atiyah at cf. In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves.
Od is surjective for any n0 and an isomorphism for n 1. The final section introduces the notion of stability of vector. The parabolic construction friedman, robert and morgan, john w. Rank two vector bundles on elliptic curves tcd maths home. Construction of the elliptic solutions r b of the aybe 78 14. Introduction vector valued modular forms have played a role in number theory 4, 7, 23. If r is a field or a discrete valuation ring of residue characteristic not 2 or 3, all these vector bundles are sums of line bundles. It also aims to provide an outline of stability of vector bundles to serve as a motivation for the study of moduli spaces of vector bundles over elliptic curves. It turns out, that all other degenerations of elliptic curves are vectorbundlewild. Elliptic curve discrete logarithm problem ecdlp is the discrete logarithm problem for the group of points on an elliptic curve over a.
From vector bundles on the elliptic curve to solutions of the aybe 78 14. Grothendieck 21 considered principal holomorphic gbundles over p1, and showed that it was always possible to reduce the structure group to a cartan subgroup, i. Gof the neutral element e, the tensorvalued function r. The interest of the elliptic curve lies in the fact that it provides the first non trivial case, grothendieck 6 having shown that for a rational curve every vector bundle. In 1957, sir michael atiyah showed that every indecomposable bundle over a smooth elliptic curve was determined by a point on the curve, and two invariants. Vector bundles on elliptic curves and factors of automorphy. Given a geometrically irreducible smooth projective curve of genus 1 defined over the field of real numbers, and a pair of integers r. Explicitly representing vector bundles over elliptic curves. Vector bundles on degenerations of elliptic curves of types. In section 5 we construct a stable vector bundle e of degree zero over a curve of genus 2, whose ring re is of smaller dimension than the group scheme.
2 1254 1504 331 289 917 1434 1111 1532 926 878 662 867 1079 517 951 879 425 428 553 702 1500 213 1198 415 187 821 1367 575 333 1036 280 1170 698 47 1028 864