Essential dimensions of isogenies

Ziquan Zhuang

Published 2024-02-23, updated 2024-03-01Version 2

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