Prym Varieties I

is a double covering, where C and C are nonsingular complete curves with Jacobians J and 3. The involution 1: C C interchanging sheets extends to t: I J, and up to some points of order two, 3 splits into an even part J and an odd part P, the Prym variety. The Prym P has a natural polarization on it, but only in two cases where 21 has zero or two branch points do we get a unique principal polarization on P, hence a theta divisor 1-. 7 c P. This is discussed in the first part of this paper (Sections 1-3). The surprise comes, however, on a closer analysis of the relations between the theta divisors 0 c J and O c 3: It turns out that they are related in a much tighter way than would be expected from looking only at the configuration of Abelian varieties and homomorphisms present. In the case of zero or two branch points this leads finally to identities relating (J, 0) and (P, F.) discovered by Schottky and Jung [15] (cf. also Riemann [13] and Farkas and Rauch [3]). The point is that the existence of any (P, °) standing in this relation to (J, 0) means that if g c 4, J is not the most general Abelian variety of dimension g! Unfortunately, an efficient method of translating this into an equivalent polynomial identity on the theta nulls of J is only known at present for g = 4. These matters are discussed in the second part of this paper (Sections 4 and 5). In the other direction, the curves C and C and their geometry can be used to compute things about P. The importance of this is that it is usually quite hard to make detailed computations on the geometry of the theta divisor in a general principally polarized n-dimensional Abelian variety [which has -n(n + 1) moduli]; those which are Jacobians of curves of genus n (with 3n — 3 moduli) are much better understood. However, by taking the Pryms for unramified double coverings C C, genus