Half a century ago Manin proved a uniform version of Serreās celebrated result on the openness of the Galois image in the automorphisms of the `$\ell$`

-adic Tate module of any non-CM elliptic curve over a given number field. In a collaboration with D. Ramakrishnan we provide first evidence in higher dimension. Namely, we establish a uniform irreducibility of Galois acting on the `$\ell$`

-primary part of principally polarized Abelian `$3$`

-folds of Picard type without CM factors, under some technical condition which is void in the semi-stable case. A key part of the argument is representation theoretic and relies on known cases of the Gan-Gross-Prasad Conjectures.

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Last updated: 12 Apr 2024