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Vector Bundles Over an Elliptic Curve
Author(s) -
Atiyah M. F.
Publication year - 1957
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-7.1.414
Subject(s) - citation , vector bundle , mathematics , elliptic curve , information retrieval , computer science , algebra over a field , discrete mathematics , library science , pure mathematics
THE primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve (defined over an algebraically closed field k). The interest of the elliptic curve lies in the fact that it provides the first nontrivial case, Grothendieck (6) having shown that for a rational curve every vector bundle is a direct sum of line-bundles. In order to provide the necessary background a certain amount of general material, not found in the literature, has been included. This consists of a brief discussion of 'Theorems A and B' and their relation with Universal bundles, a little on protective bundles, and some results on reduction of structure group. The case of vector bundles over an algebraic curve is treated in greater detail, and more precise results are obtained. In particular a refinement of Theorems A and B is given (Theorem 1) which seems to be a necessary preliminary in any attempt at classification of vector bundles. This concludes Part I of the paper. Part II is devoted to the classification of vector bundles over an elliptic curve. The problem is completely solved and the main result is stated in Theorem 7. The characteristic of the field does not enter into this part of the problem, and the results are valid in both characteristic 0 and p. In Part III we examine the operation of the tensor product. This is most easily expressed in terms of the ring S generated by the vector bundles (cf. Part I, § 1). We show (Theorem 12) that $ is the tensor product of certain sub-rings SQ and Sp (over all primes p), and the ring structure of SQ and Sp is given by Theorems 8, 13, and 14. These results are all for the characteristic zero case, and we make only a few isolated remarks for the case of characteristic p. We conclude the paper with a few brief applications of the results of Parts II and III.