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See detailBig monodromy theorem for abelian varieties over finitely generated fields
Arias De Reyna Dominguez, Sara UL; Gajda, Wojciech; Sebastian, Petersen

in Journal of Pure and Applied Algebra (2013), 217(2), 218--229

An abelian variety over a field K is said to have big monodromy, if the image of the Galois representation on l-torsion points, for almost all primes l contains the full symplectic group. We prove that ... [more ▼]

An abelian variety over a field K is said to have big monodromy, if the image of the Galois representation on l-torsion points, for almost all primes l contains the full symplectic group. We prove that all abelian varieties over a finitely generated field K with the endomorphism ring Z and semistable reduction of toric dimension one at a place of the base field K have big monodromy. We make no assumption on the transcendence degree or on the characteristic of K. This generalizes a recent result of Chris Hall. [less ▲]

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