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Bulletin of the London Mathematical Society 1994 26(5):417-421; doi:10.1112/blms/26.5.417
© 1994 by London Mathematical Society
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© London Mathematical Society

The Tate Conjecture for Generic Abelian Varieties

Salman Abdulali

Department of Mathematics University of Toronto Toronto Canada M5S 1A1

Received 16 June 1992. Revision received 8 April 1993.

Let A -> V be a Kuga fibre variety of Mumford's Hodge type, defined over a finitely generated subfield of C, and let {eta} be the generic point of V. We show that any element of Formula which is invariant under Formula, for some finite extension E of k({eta}), is fixed by the semisimple part of the Hodge group of A{eta}. If A -> V satisfies the H2-condition, then the Hodge and Tate conjectures are equivalent for A{eta}, and the Mumford–Tate conjecture is true.


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