Let $K$ be a number field. The Frey-Mazur conjecture asserts the existence of a constant $N_K$ such that for all primes $p>N_K$, and all pairs of elliptic curves $E_1$, $E_2/K$, if $\overline{\rho}_{E_1,p} \sim \overline{\rho}_{E_2,p}$ then $E_1$ is isogenous to $E_2$.

My question is about generalizations of this conjecture to higher dimensional abelian varieties. Has such a conjecture been formulated? Or are there subtleties present in the higher dimensional setting that would lead us to expect that such a generalization is false?


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