# Abstract' homomorphisms of l-adic Galois groups and Abelian varieties

Mathematics – Number Theory

Scientific paper

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## Details Abstract' homomorphisms of l-adic Galois groups and Abelian varieties Abstract' homomorphisms of l-adic Galois groups and Abelian varieties

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Scientific paper

Let $k$ be a totally real field, and let $A/k$ be an absolutely irreducible, polarized Abelian variety of odd, prime dimension whose endomorphisms are all defined over $k$. Then the only strictly compatible families of abstract, absolutely irreducible representations of $\gal(\ov{k}/k)$ coming from $A$ are tensor products of Tate twists of symmetric powers of two-dimensional $\lambda$-adic representations plus field automorphisms. The main ingredients of the proofs are the work of Borel and Tits on the abstract' homomorphisms of almost simple algebraic groups, plus the work of Shimura on the fields of moduli of Abelian varieties.

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