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We give a formula for the essential dimensions of isogenies A A' of complex abelian varieties when the degree of the isogeny is coprime to (A) !. As a corollary, we show that the p-multiplication map (where p is a prime) on an abelian variety is incompressible when p> A or when A 4. This confirms a conjecture of Brosnan for most primes.
Ziquan Zhuang (Fri,) studied this question.