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

Abstract

Let $C$ be an elliptic curve defined over $\mathbb{Q}$. We can associate two formal groups with $C$: the formal group $\^{C}(X,Y)$ determined by the formal completion of the Néron model of $C$ over $\mathbb{Z}$ along the zero section, and the formal group $F_L(X,Y)$ of the L-series attached to $l$-adic representations on $C$ of the absolute Galois group of $\mathbb{Q}$. Honda shows that $F_L(X, Y)$ is defined over $\mathbb{Z}$, and it is strongly isomorphic over $\mathbb{Z}$ to $\^{C}(X,Y)$. In this paper we give a generalization of the result of Honda to building blocks over finite abelian extensions of $\mathbb{Q}$. 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 $\rm{GL}_2$-type. It also contains the generalization of our previous paper to $\mathbb{Q}$-curves over quadratic fields.