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

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

2001-06-03

arxiv.org/abs/math/0106016v2

Mathematics

Number Theory

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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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