Clifford Algebras and Families of Abelian Varieties

Abstract

In the arithmetic theory of automorphic functions on a symmetric bounded domain=G/K, as developed recently by Shimura and Kuga [2], [2a], it is important to consider a family of (polarized) abelian varieties onobtained from a symplectic representationρ(defined over Q) ofG(viewed as an algebraic group defined over Q) satisfying a certain analyticity condition. Recently, I have determined completely such representations, reducing the problem to the case whereGis a Q-simple group and whereρis a Q-primary representation ([3], [4]). It has turned out that, besides the four standard solutions investigated already by Shimura, there exist two more non-standard solutions, one of which comes from a spin representation of the orthogonal group and thus gives a family of abelian varieties on a domain of type (IV). The purpose of this short note is to explain how one can construct most simply, starting from the “regular representation” of the corresponding Clifford algebra, examples of such families, including also the non-analytic case.