Formal Groups of Building Blocks Completely Defined Over Finite Abelian Extensions of Q

Let C be an elliptic curve defined over Q. We can associatetwo formal groups with C: the formal group (X, Y) determinedby the formal completion of the Néron model of C overZ along the zero section, and the formal group F_L(X, Y) of theL-series attached to l-adic representations on C of the absoluteGalois group of Q. Honda shows that F_L(X, Y) is defined overZ, and it is strongly isomorphic over Z to (X, Y). In this paperwe give a generalization of the result of Honda to buildingblocks over finite abelian extensions of Q. The difficulty isto define new matrix L-series of building blocks. Our generalizationcontains the generalization of Deninger and Nart to abelianvarieties of GL₂-type. It also contains the generalization ofour previous paper to Q-curves over quadratic fields. 2000 MathematicsSubject Classification 11G10 (primary), 11F11 (secondary).