An abelian variety B/Q is called an abelian Q-variety if for each σ ∈ G_Q = Gal(Q/Q) there exists an isogeny μ_σ: ~σB→B compatible with the endomorphisms of B, i.e. such that φ o μ_σ o ~σφ for all φ∈ End0/Q(B) = End/Q(B) ? z Q. A building block is an abelian Q-variety B whose endomorphism algebra End0/Q(B) is a central division algebra over a totally real number field F with Schur index t = 1 or t = 2 and t[F : Q] = dim B. In the case t = 2 the quaternion algebra is necessarily totally indefinite. The interest in the study of the building blocks comes from the fact that they are the absolutely simple factors up to isogeny of the non-CM abelian varieties of GL2-type (see [Py]) and therefore, as a consequence of a generalization of Shimura-Taniyama, they are the non-CM absolutely simple factors of the modular jacobians J_1(N).
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