On Products and Algebraic Families of Jacobian Varieties

Abstract

1. The first theorem of this paper (? 2) is an extension to products of jacobian varieties of Matsusaka's characterization of a jacobian variety [6, Th. 3]. As a corollary (? 3) we obtain a result which was proved in the author's thesis (Chicago, 1958) and which is cited by Baily in the proof of the following: Let E denote the set of points corresponding to period matrices for Riemann surfaces of genus n in the quotient space VE = H,, / F,,. Then E is an irreducible locally closed subset of Vn in the Zariski topology determined on V. by the modular forms for L'n on Hn. [1, Cor. to Th., p. 313.] Our second theorem (? 5) is an extension of this result of Baily to algebraic families of polarized abelian varities (Weil's generalizations of the spaces Vs)'. The author wishes to express his appreciation to Professor Matsusaka for his advice and encouragement.