Formal groups and Dirichlet L-functions, II

Abstract

In [4], Honda presents a number of results concerning formal groups constructed from Dirichlet series. In particular, he shows that the formal group associated to the L-series for a quadratic character mod D is integrally isomorphic to the algebroid formal group X+ Y+ fi XY (Theorem 2 below). In this paper we generalize this result to non-quadratic characters by using higher dimensional analogues of Honda’s groups. In [S] Honda uses his result to give a proof of quadratic reciprocity. One of our aims in generalizing Honda’s result is to prove similarly the reciprocity law in a cyclotomic field. Replacing the algebroid group by the formal group 8 of an elliptic curve 6’ over Q, Honda also shows in [4] that 8 is strictly isomorphic to the formal group obtained from the zeta function of 8. (See also [3 1.) Moreover, this isomorphism is integral, i.e., its coefficients are in E. In this case, the analogue of quadratic reciprocity is the Atkin Swinnerton-Dyer congruences. As usual, for a prime p we will let Z,, Q,, and CP denote the p-adic integers, the p-adic rationals, and the completion of an algebraic closure of Q,, respectively. Also ord, will denote the p-adic valuation on C, normalized so that ord,(p) = 1. The authors thank Robert Gold and Lynne Walling for helpful conversations.