If k is the complex field then it has been shown by serre 9 that the algebraic and. Od is surjective for any n0 and an isomorphism for n 1. Coherent state transforms and vector bundles on elliptic. In the case of a projective algebraic variety the converse is also true. Given a geometrically irreducible smooth projective curve of genus 1 defined over the field of real numbers, and a pair of integers r. Holomorphic principal bundles over elliptic curves ii. This chapter gives a variety of techniques for understanding stable rank 2 vector bundles v over elliptic surfaces x. Irreducible vector bundles on some elliptic nonkahler. Vector bundles on degenerations of elliptic curves of. Explicitly representing vector bundles over elliptic curves. The author was partially supported by murst and gnsaga ofcnr italy. Vector bundles on elliptic curves and factors of automorphy.
The simplest example is known as the trivial vector bundle of rank n, i. In 1957, sir michael atiyah showed that every indecomposable bundle over a smooth elliptic curve was determined by a point on the curve, and. 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. Limit linear series for vector bundles teixidor i bigas, montserrat, tohoku mathematical journal, 2014. Moduli spaces of hecke modi cations for rational and. Vector bundles and torsion free sheaves on degenerations. It allows us to study degenerations of solutions of the classical yangbaxter equa.
Coherent state transforms and vector bundles on elliptic curves. This course note aims to give a basic overview of some of the main lines of study of elliptic curves, building on the students knowledge of undergraduate algebra and complex analysis, and filling in background material where required especially in number theory and geometry. The parabolic construction friedman, robert and morgan, john w. The subbundles of decomposable vector bundles over an.
Pdf coherent state transforms and vector bundles on. Interpolation and vector bundles on curves abstract interpolation is a property of vector bundles on curves closely related to slope stability. In section 2 we will prove that they trivialize over a principal gbundle with. We shall be concerned with vector bundles over x, i. Seminar on 2vector bundles and elliptic cohomology, v. Rn ian dundas and john rognes dedicated to graeme segal on the occasion of his 60th birthday 1. Vector bundles on degenerations of elliptic curves and yangbaxter equations igor burban and bernd kreussler abstract. We also recall basic facts about hilbert scheme of points of smooth complex surfaces and explain the relation with the moduli space of higgs. 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. In this paper we introduce the notion of a geometric associative rmatrix attached to a genus one.
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. An introduction to the theory of elliptic curves pdf 104p covered topics are. The problem of classification of vector bundles over an elliptic curve i. Introduction the work to be presented in this paper has been inspired by several of professor graeme segals papers. A slope of a holomorphic vector bundle w over a nonsingular algebraic curve or over a riemann surface is a rational number. Vector bundles over elliptic surfaces springerlink. Fouriermukai transforms and stable bundles on elliptic curves georg hein and david ploog ebruaryf 14, 2005 abstract ew prove atiyahs classi cation results about indecomposable vector bundles on an elliptic curve by applying the ouriermukf ai transform. Elliptic curves, the geometry of elliptic curves, the algebra of elliptic curves, elliptic curves over finite fields, the elliptic curve discrete logarithm problem, height functions, canonical heights on elliptic curves, factorization using elliptic curves, lseries. Rank 3 lagrangian vector bundles on an elliptic curve. Vector bundles on elliptic curves over a discrete valuation ring e. Elliptic curves over the complex numbers are precisely the same thing as riemann surfaces of genus 1, hence the same as 1loop string diagrams.
Using fouriermukai transforms, we prove some results about the. We study vector bundles on the moduli stack of elliptic curves over a local ring r. The subbundles of decomposable vector bundles over an elliptic curve 187 with a polystable bundles e on c and we obtain bundles on x making elementary transformations of e. Elliptic curves can have points with coordinates in any. View enhanced pdf access article on wiley online library html view download pdf for offline viewing.
For r z 3, we construct higher rank indecomposable vector bundles and give a classification of vector bundles that are iterated extensions of line bundles. Coherent state transforms and vector bundles on elliptic curves carlos a. Varieties of small degree with respect to codimension and ample vector bundles lanteri, antonio and novelli, carla, journal of the mathematical society of japan, 2008. Rank two vector bundles on elliptic curves 115 example 2. Vector bundles and torsion free sheaves on degenerations of elliptic curves 3 of indecomposable maximal cohenmacaulay modules follows from atiyahs classi. Free elliptic curves books download ebooks online textbooks. We prove atiyahs classification results about indecomposable vector bundles on an elliptic curve by applying the fouriermukai transform. These factors of automorphy can be obtained from results in 14 and checked by using wellknown results regarding vector bundles on elliptic curves. It allows us to study degenerations of solutions of the classical yangbaxter. Vector bundles on the moduli stack of elliptic curves.
Simple vector bundles on plane degenerations of an elliptic curve lesya bodnarchuk, yuriy drozd, and gertmartin greuel abstract. In cases of elliptic curves and projective configurations of. Simple vector bundles on plane degenerations of an elliptic curve 21 in most situations it is useful to present a representation m as a blockmatrix with the block mx on the place ij for x g qoj,i as in the previous sections, with a little abuse of notations, we write the matrices m and 5 in a form of a table with x on the ijentry. Vector bundles of degree zero over an elliptic curve, flat bundles. Full text of simple vector bundles on plane degenerations.
Our description of simple vector bundles on degenerate elliptic curves contributes to a better understanding of the theory of vector bundles on elliptic. Transactions of the american mathematical society volume 364, number 1, january 2012, pages 7174 s 000299472011053547 article electronically published on august 25, 2011. On the ring of unipotent vector bundles on elliptic curves in positive characteristics stefan schroer revised version, 8 february 2010 abstract. Vector bundles over an elliptic curve 415 embedded biregularly in some projective space. Since it is known that onedimensional complex tori correspond to elliptic curves and since the classi. Springer new york berlin heidelberg hong kong london milan paris tokyo.
Bundles on elliptic curves in this section we show that an indecomposable bundle on an elliptic curve is ample if and only if its degree is positive. Every algebraic subvariety of an algebraic variety over cde. In this case, we prove that every rank2 irreducible vector bundle on x is a pullback from b up to a twist by a line bundle. Part v of our seminar on elliptic cohomology and 2vector bundles. G sln,c or g p gln,c, without however considering the problem of.
However, they turn out to be such in some bigger class of matrix problems. Elliptic curve discrete logarithm problem ecdlp is the discrete logarithm problem for the group of points on an elliptic curve over a. This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. In part 1 we consider the family of vector bundles of degree zero over an elliptic curve. We translate the atiyahs results on classification of vector bundles on elliptic curves to the language of factors of automorphy. Fouriermukai transforms and stable bundles on elliptic curves. 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. Vector bundles on degenerations of elliptic curves and. From vector bundles on the elliptic curve to solutions of the aybe 78 14. Rank two vector bundles on elliptic curves tcd maths home. For simplicity, we shall assume that x is simply connected, so that the base. Effective divisor classes of a projective plane bundle over an elliptic curve takahashi, tomokuni, rocky mountain journal of mathematics, 2019. The notion is motivated by the deformation theory of curves in projective space incident to given. Construction of the elliptic solutions r b of the aybe 78 14.
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