Published in Contemporary Mathematics, Volume 133, January 1, 1992, pages 141-162.
After a survey of the Weierstrass family and cohomology, we compute the lifted homology of the Weierstrass family with compact supports so that explicit formulae for the zeta function of each fibre of the Weierstrass family may be obtained. The (co-)homology theory that we use is found in [L1], [L2] and [L3]. Therefore, this article can be regarded as an application of Lubkin's p-adic theory of cohomologies to an algebraic family called the Weierstrass scheme over the ring (Z/pZ)[g2,g3). The cohomological background for the computation will be rather carefully exploited.