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Relative splitting of one-motives. (English) Zbl 1043.11054

Murty, V. Kumar (ed.) et al., Number theory. Proceedings of the international conference on discrete mathematics and number theory, Tiruchirapalli, India, January 3–6, 1996 on the occasion of the 10th anniversary of the Ramanujan Mathematical Society. Providence, RI: American Mathematical Society (ISBN 0-8218-0606-8/pbk). Contemp. Math. 210, 3-17 (1998).
From the text: In a previous paper “1-motifs et relations d’orthogonalité dans les groupes de Mordell-Weil” in: Diofantovye Priblizhenya, Fel’dman volume, Mat. Zap. 2, 7–22 (1996) the author introduced two families of one-motives \(M\) over a number field \(k\): those whose associated Galois representations are “relatively split”, and those with “relative good reduction at all places of \(k\)”. We showed that the first family is included in the second one, and conjectured that they coincide when \(k\) is totally real. In the first two parts of the present paper, we (hope to) shed new light on these notions by:
i) defining the elements \(M\) of the first family in terms of the weight filtration of their Mumford-Tate group \(P(M)\), viz.: the subgroup \(W_{-2} (P(M))\) of \(\operatorname{Hom}(X\otimes Y,\mathbb{Q}(1))\) induced by \(M\) vanishes.
ii) characterizing the second family through the category \(MM(S)\) (introduced in [10]) of mixed motives over \(S\), viz.: the class \(c(M)\) provided by \(M\) in \(\text{Ext}^2_{MM(S)}(X\otimes Y(0)\), \(\mathbb{Q}(1))\) vanishes.
Here, \(X\) and \(Y\) are constant groups attached to the one-motive \(M\), and \(S\) is the ‘compactification’ of the ring of integers of \(k\).
The conjectural implication (ii)\(\to\)(i) for totally real \(k\)’s is related to a natural problem in transcendence theory. While we make no progress on the case relevant to the conjecture, we do prove in the last section of the paper an analogous result for vector group extensions of abelian varieties with real multiplications.
For the entire collection see [Zbl 0878.00049].

MSC:

11G09 Drinfel’d modules; higher-dimensional motives, etc.
11G35 Varieties over global fields
14G25 Global ground fields in algebraic geometry
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