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Multiplicative isogeny estimates

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  09 April 2009

David Masser
David Masser
Affiliation:
Mathematisches Institut Universität BaselRheinsprung 21 4051 BaselSwitzerland
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Abstract

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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).

MSC classification

Secondary: 11G10: Abelian varieties of dimension $> 1$ 11G05: Elliptic curves over global fields
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 2 , April 1998 , pp. 178 - 194
Copyright
Copyright © Australian Mathematical Society 1998

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