## Honda Theory for Formal Groups of Abelian Varieties over $\mathbb Q$ of $\mathbf GL_{2}$-Type

J. Math. Sci. Univ. Tokyo
Vol. 21 (2014), No. 2, Page 355–372.

Miyasaka, Yuken ; Shinjo, Hirokazu
Honda Theory for Formal Groups of Abelian Varieties over $\mathbb Q$ of $\mathbf GL_{2}$-Type
Honda proved that two formal groups attached to an elliptic curve $E$ over $\mathbb Q$ are strongly isomorphic over $\mathbb Z$, where one of them is obtained from the formal completion along the zero section of the Néron model over $\mathbb Z$ and another is obtained from the L-series attached to the $l$-adic Galois representations on $E$. In this paper, we generalize his theorem to abelian varieties over $\mathbb Q$ of $\mathbf GL_{2}$-type. As an application, we give a method to calculate the coefficients of the L-series attached to an algebraic curve over $\mathbb Q$ with a Jacobian variety of $\mathbf GL_{2}$-type.
Keywords: Formal group, abelian variety of $\mathbf GL_{2}$-type, complex multiplication, $\mathbb L$-series.