Cohomological realization of a family of 1-motives

• Published in: Journal of number theory Vol. 25; no. 2; pp. 152 - 161
• Main Author: Ribet, Kenneth A.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.02.1987
• ISSN: 0022-314X; 1096-1658
• DOI: 10.1016/0022-314X(87)90021-7

The author studies a class of 1-motives which he encountered in his work joint work with Jacquinot ( J. Number Theory 24, No. 3, 1986). A 1-motive M was attached in this previous work to each triple ( A, f, α), where A is an abelian variety, f: A → A′ a homomorphism from A to its dual, and α a point on A. It was stated, without proof, at that time that the 1-motive M generates the same “Galois groups” (in a literal sense if one discusses n-torsion points of 1-motives, or else in the sense of Mumford-Tate groups if one works over C) as the simpler 1-motive M = [ Z → A] which “is” α. The present article refines that observation by proving that the cohomological realizations of M may be derived from those of M , and vice versa, by linear algebra constructions. More precisely, let T( M) and T( M ) be the modules of “ n-division points” of M and M , where n ≥ 1 is an integer. The latter module is an evident quotient of the former. The author shows that, in the other direction, T( M) may be recovered from T( M ) as a specific subquotient of T( M ) ⊗ 2.