STNB2025 (38th edition)
Abelian varieties genuinely of GL(n)-type
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Abstract
In this talk, I will explain the properties of abelian varieties (genuinely) of $GL_n$-type, and in particular those which are geometrically of the first kind. This gives a generalization of Ribet's theory of abelian varieties of $GL_2$-type without potential complex multiplication. I will introduce building blocks, inner twists, and the attached system of Galois representations. With the mentioned hypotheses, we achieve images of Galois in $GSp_n$ and $GO_n$, with similitude factor given by a certain nebentype. I will also showcase a family of abelian fourfolds genuinely of $GL_4$-type whose attached Galois representations are symplectic. This is joint work with Francesc Fité and Xavier Guitart.
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