Formal groups and zeta-functions of elliptic curves

Abstract

The author shows that the isomorphism class of a formal group overZ/pZ (resp. overZ p ) of finite height (resp. having reduction modp of finite height) is determined by its characteristic polynomial. It is then proved that the formal groups associated to a large class of Dirichlet series with integer coefficients are defined overZ. Finally, these results are used to extend a theorem of Honda (Osaka J. Math.5, 199–213 (1968), Theorem 5) to include the case of supersingular reduction at the primes 2 and 3. LetE be an elliptic curve defined overQ, andF(x, y) be a formal minimal model forE. LetG(x, y) be the formal group associated to the globalL-seriesL(E, s) ofE overQ. Honda's theorem now becomes:G(x, y) is defined over Z and is isomorphic over Z to F(x, y).