MULTIPLICATIVE ISOGENY ESTIMATES

The theory of isogeny estimates for Abelian varieties provides 'additi ve bounds' of the form 'd is at most B' for the degrees d of certain i sogenies. We investigate whether these can be improved to 'multiplicat ive bounds' of the form 'a' divides B'. We find that in general the an swer is no (Theorem 1), but that sometimes the answer is yes (Theorem 2). Further we apply the affirmative result to the study of exceptiona l primes l in connexion with modular Galois representations coming fro m elliptic curves: we prove that the additive bounds for l of Masser a nd Wustholz (1993) can be improved to multiplicative bounds (Theorem 3 ).