We prove here that the p - 1 first derivatives of the fundamental period of the Carliz module are algebraically independent. For that purpose we will show to use Mahler's method in this situtaion.

The theory of isogeny estimates for Abelian varieties provides ‘additive bounds’ of the form ‘d is at most B’ for the degrees d of certain isogenies. We investigate whether these can be improved to ‘multiplicative bounds’ of the form ‘d divides B’. We find that in general the answer is no (Theorem 1), but that sometimes the answer is yes (Theorem 2). Further we apply the affirmative result to the study of exceptional primes ℒ in connexion with modular Galois representations coming from elliptic curves: we prove that the additive bounds for ℒ of Masser and Wüstholz (1993) can be improved to multiplicative bounds (Theorem 3).

Let F(X, Y) be an absolutely irreducible polynomial with coefficients in an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X, Y) = 0 and by K[C] the ring of regular functions on Cover K. Assume that there is a unit ϕ in K[C] − K such that 1 − ϕ is also a unit. Then we establish an explicit upper bound for the size of integral solutions of the equation F(X, Y) = 0, defined over K. Using this result we establish improved explicit upper bounds on the size of integral solutions to the equations defining non-singular affine curves of genus zero, with at least three points at ‘infinity’, the elliptic equations and a class of equations containing the Thue curves.