FORMAL GROUPS OF BUILDING BLOCKS COMPLETELY DEFINED OVER FINITE ABELIAN EXTENSIONS OF $\mathbb{Q}$

Abstract

Let C be an elliptic curve defined over Q. We can associate two formal groups with C: the formal group Ĉ(X, Y) determined by the formal completion of the Néron model of C over Z along the zero section, and the formal group FL(X, Y) of the L-series attached to l-adic representations on C of the absolute Galois group of Q. Honda shows that FL(X, Y) is defined over Z, and it is strongly isomorphic over Z to Ĉ(X, Y). In this paper we give a generalization of the result of Honda to building blocks over finite abelian extensions of Q. The difficulty is to define new matrix L-series of building blocks. Our generalization contains the generalization of Deninger and Nart to abelian varieties of GL2-type. It also contains the generalization of our previous paper to Q-curves over quadratic fields. 2000 Mathematics Subject Classification 11G10 (primary), 11F11 (secondary).