Abelian varieties genuinely of GL_n-type get a theory of building blocks, inner twists and nebentypes, with a proof that their λ-adic representations are symplectic or orthogonal whenever the base-changed endomorphism center is totally real, plus explicit abelian fourfolds.
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Abelian varieties genuinely of $\mathrm{GL}_n$-type
Abelian varieties genuinely of GL_n-type get a theory of building blocks, inner twists and nebentypes, with a proof that their λ-adic representations are symplectic or orthogonal whenever the base-changed endomorphism center is totally real, plus explicit abelian fourfolds.