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It is proved that every morphism of a curve with an involution into an Abelian variety, anticommuting with the involution, factors through the associated Prym variety. This result is used to show that Jacobians of curves of genus five arise as Prym varieties associated to a certain class of curves. 1. Principally polarized Abelian varieties. Let A' be a complex torus. We have X — E\V, where E «« Cg is the universal covering space of Xand V is a lattice in F. Then tt^X) = HxiX,Z)=V and H*iX, Z) = Hom(AK, Z); where A V is the exterior algebra generated by V over Z. Let S be a theta-divisor on X. 2 is called principal if the complete linear series of 2 is one-dimensional, i.e., if 121 = dim H°iOx(Z)) = 1. X together with a principal theta-divisor 2 is called a principally polarized Abelian variety. X = ?ic°iX) is called the dual variety. A polarization 2 of X induces a homomorphism X£ : X —► X defined by Xr(;c) = 7^(2) 2, where Tx is the translation by x. It is known that if 2 is principal, then X2 is an isomorphism and 2 X£(2) is a principal theta-divisor on X [Ml]. There exists a positive definite Hermitian form // = 4> + i\p on F such that the restriction of i// to V A Fis integer-valued and a map a: V—*■ S1, the unit circle, such that aiux + u2) = a(«x)a(M2) exp mi>iux, u2) with the property that 2 is the divisor of zeros at the theta function 0(z) with the periodicity relations 0(z + u) = a(«)exp tt(//(z, u) + !4/Y(u, u))diz) for all zEE, u E F (ibid.). Note that tf> E Hom(A2K, Z) = //2(X, Z). It is known from topology that \p is the Poincaré dual of 2, i.e., \¡i n iX) = [2] EH2 _2(X, Z), where iX) is the orientation class of X and [2] is the homology class represented by Received by the editors May S, 197S. AMS (MOS) subject classifications (1970). Primary 14H30, 14K99, 30AS2; Secondary 14H40, 30A48.

Universal properties of Prym varieties with an application to algebraic curves of genus five