Formal groups of Q-curves with complex multiplication
by
Fumio Sairaiji
Hiroshima International University

Let E be an elliptic curve defined over Q. We can associate two formal groups with E, the formal group F_E(X, Y) determined by the formal completion of the Néron model of E over Z along the zero section and the formal group F_L(X, Y) of the L-series attached to l-adic representations on E of the absolute Galois group of Q. T. Honda shows that F_L(X, Y) is defined over Z, and it is strongly isomorphic over Z to F_E(X, Y). The author generalizes it to building blocks over finite abelian extensions of Q, after the generalization by Deninger-Nart to abelian varieties of GL₂-type.

In this talk we give a generalization of the result of Honda to Q-curves with complex multiplication by an imaginary quadratic field. Such Q-curves are classified by T. Nakamura. The difference from the previous genelarizations is that such Q-curve is not always defined over abelian extensions of Q.

Date received: May 18, 2008