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The Tate Conjecture for Generic Abelian Varieties
Author(s) -
Abdulali Salman
Publication year - 1994
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/26.5.417
Subject(s) - mathematics , conjecture , abelian group , pure mathematics , invariant (physics) , abelian variety , variety (cybernetics) , finitely generated abelian group , hodge theory , discrete mathematics , combinatorics , cohomology , statistics , mathematical physics
Let A → V be a Kuga fibre variety of Mumford's Hodge type, defined over a finitely generated subfield of C, and let η be the generic point of V . We show that any element ofH e ´ t 2 r( A η ¯ , Q l ) ( r )which is invariant under Gal (k ( η ) ¯ / E ) , for some finite extension E of k ( η ), is fixed by the semisimple part of the Hodge group of A η . If A → V satisfies the H 2 ‐condition, then the Hodge and Tate conjectures are equivalent for A η , and the Mumford–Tate conjecture is true.
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