Semistable Bundles over an Elliptic Curve

Abstract We identify the moduli space M n, d of semistable bundles of rank n and degree d on an elliptic curve C as a symmetric product of the curve, its dimension being the greatest common divisor of n and d. Under this identification the determinant map det: M n, d corresponds to the Abel-Jacobi map α: ShC → Jh, so that the fixed-determinant moduli space is identified with a projective space. We then describe the Brill-Noether loci, theta divisors, and Picard groups and compute the space of theta functions for both the full moduli space and the fixed-determinant moduli space of any rank and any degree. We extend to elliptic curves a theorem of Drezet and Narasimhan on the Picard group of moduli spaces, as well as two formuulas of Beauville, Narasimhan, and Ramanan on theta functions of level one. Our results verify in particular for an elliptic curve a formula of E. Verlinde, and an equality first noticed by Bott and Szenes relating theta functions over the full moduli space and over the fixed-determinant moduli space.