A Property of the Periods of a Prym Differential

The periods of Prym differentials can be used to prove the invariance of Picard bundles on Jacobian varieties. Let S be a compact Riemann surface of genus g with universal covering surface 77 with the deck transformation group D. For any homomorphism X: D -> C* = C {0} into the group of complex units, a (meromorphic) Prym differential on S1 with multipliers A1 is a meromorphic differential w on 77, satisfying the condition w(du) = X(d)w(u) for all din D and u in 77. Let p be a fixed point of U. For all X, consider the space of Prym differentials which are regular except for a pole of at most order one at p or its translates under D. This space has dimension g and varies complex analytically as one varies X (see below). These spaces form a vector bundle over the space of all X. In this paper, I show how to trivialize this vector bundle over a g dimensional real torus, consisting of special A"s, by means of the periods of Prym differentials. In a previous paper [3], I proved that such a trivialization was possible by abstract reasoning. The methods of this paper are more elementary and yield an explicit trivialization. Some basic facts about this vector bundle can be found in [2]. A general reference for periods of Prym differentials is the book [1]. Given any Prym differential w, let V be the largest open subset of 77 such that w has zero residues everywhere in V. Denote the image of V in S by 7. Let v be any fixed point of V and let t be its image in 7. For any path a in 7, beginning and ending at t, the a period Aa of w is defined by the integral Aa= -X(a)-lJ,w, where a is the unique path in U, lifting a, which begins at t and ends at a ■ t. The deck transformation a will henceforth be denoted by a. Thus, A0 depends only on the homotopy class of a in the fundamental group of 7. Furthermore, Aa is defined for most a in any homotopy class. We can then regard Aa as a function on 77,(7, i). Also, it satisfies the cocycle identity, A0T = X(t)~1A0 + AT for all a and t in w,(7, t). Received by the editors February 5, 1975. AMS (MOS) subject classifications (1970). Primary 32L05, 14K30.