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.

Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic

“…The Kuga-Satake abelian variety A associated to (X, L) is exactly the complex torus C + (V R )/C + (L) where C + (V R ) is considered as a complex vector space via the complex structure given by multiplication by e + . For further details we refer to the articles of Satake [31], Kuga and Satake [15] and van Geemen [33].…”

Periods of Hilbert Modular Surfaces

“…In §5 of Chapter IT, we recall the formalism constructing abelian varieties attached to Clifford algebras, following Satake [40], and Deligne [10]. Applying this construction to the Hodge structure H 2 (M f ,l'l), we have an abelian variety A(f) of dimension 4[K f :I'l].…”

The congruence relation in the non-PEL case

“…Remark 5.2. Using Clifford algebras in the way Satake does in [17], one can indeed show that 5-dimensional Shimura data (G, Y ) as in the theorem exist. Such a G is the spinor similitude group of a certain 7dimensional quadratic Q-space of signature (2, 5), moreover, by an easy application of Meyer's theorem the spinor similitude group of every Qform of the quadratic R-space of signature (2, 5) can be imbedded into G D , for some D that depends on the quadratic Q-space.…”