FORMAL GROUPS AND ZETA-FUNCTIONS

Abstract

Cl be an elliptic curve over the rational number field Q, uniformized by automorphic functions with respect to some congruence modular group T0(N). In the language of formal groups results of Eichler [3] and Shimura [14] imply that a formal completion C1 of Cl (as an abelian variety) is isomorphic over Z' to a formal group whose invariant differential has essentially the same coefficients as the zeta-f unction of Q. In this paper we prove that the same holds for any elliptic curve C over Q (th. 5). This follows from general theorems which allow us explicit construction and characterization of certain important (one-parameter) formal groups over finite fields, p-adic integer rings, and the rational integer ring (th. 2 and th. 3). The proof of th. 5 depends only on the fact that the Frobenius endomorphism of an elliptic curve over a finite field is the inverse of a zero of the numerator of the zeta-function, and implies a general relation between the group law and the zetafunction of a commutative group variety. In fact it is remarkable that the p-f actor of the zeta-function of C for bad p also can be given a clear interpretation from our point of view (cf. th. 5). Moreover, we prove that the Dirichlet L-function with conductor D has the same coefficients as the canonical invariant differential on a formal group isomorphic, over the ring of integers in Q(\/ D), to the algebroid group x+y+\/T) xy (th. 4). In this way the zeta-function of a commutative group variety may be characterized as the L-series whose coefficients give a normal form of its group law.