of algebraic curves, write Ji for the Jacobian variety of Ci. Then from φ construct the corresponding morphism
ψ: J1 → J2,
which can be defined on a divisor class D of degree zero by applying φ to each point of the divisor. This is a well-defined morphism, often called the norm homomorphism. Then the Prym variety of φ is the kernel of ψ. To qualify that somewhat, to get an abelian variety, the connected component of the identity of the reduced scheme underlying the kernel may be intended. Or in other words take the largest abelian subvariety of J1 on which ψ is trivial.
The theory of Prym varieties was dormant for a long time, until revived by David Mumford around 1970. It now plays a substantial role in some contemporary theories, for example of the Kadomtsev–Petviashvili equation. One advantage of the method is that it allows one to apply the theory of curves to the study of a wider class of abelian varieties than Jacobians. For example, principally polarized abelian varieties (p.p.a.v.'s) of dimension > 3 are not generally Jacobians, but all p.p.a.v.'s of dimension 5 or less are Prym varieties. It is for this reason that p.p.a.v.'s are fairly well understood up to dimension 5.
References
Birkenhake, Christina; Lange, Herbert (2004). "Prym varieties". Complex Abelian Varieties. New York: Springer-Verlag. pp. 363–410. ISBN3-540-20488-1.
Mumford, David (1974), "Prym varieties. I", in Ahlfors, Lars V.; Kra, Irwin; Nirenberg, Louis; et al. (eds.), Contributions to analysis (a collection of papers dedicated to Lipman Bers), Boston, MA: Academic Press, pp. 325–350, ISBN978-0-12-044850-0, MR0379510