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Arithmetic duality theorems for 1-motives. (English) Zbl 1088.14012

J. Reine Angew. Math. 578, 93-128 (2005); corrigendum ibid. 632, 233-236 (2009).
Let \(F\) be a field. A \(1\)-motive over \(F\) is defined to be a two-term complex \(M\) of \(F\)-group schemes \([Y\to G]\) (placed in degrees \(-1\) and \(0\)), where \(Y\) is the \(F\)-group scheme associated to a finitely generated free abelian group equipped with a continuous \(\text{Gal}(F)\)-action and \(G\) is semi-abelian variety over \(F\) (an extension of an abelian variety \(A\) by a torus \(T\)). A \(1\)-motive \(M\) has a Cartier dual \(M^*=[Y^*\to G^*]\) equipped with a canonical pairing \(M\otimes^{\mathbb{L}} M^*\to {\mathbb{G}}_m[1]\). The main results of the paper are formulated as follows.
Theorem 1. Let \(K\) be a local field and let \(M=[Y\to G]\) be a \(1\)-motive over \(K\). For \(i=-1, 0, 1, 2\), there are canonical pairings \({\mathbb{H}}^i(K,M)\times {\mathbb{H}}^{1-i}(K,M^*)\to {\mathbb{Q}}/{\mathbb{Z}}\) inducing perfect pairings between (1) the profinite group \({\mathbb{H}}^{-1}_{\wedge}(K,M)\) and the discrete group \({\mathbb{H}}^2(K,M^*)\), and (2) the profinite group \({\mathbb{H}}^0(K,M)^{\wedge}\) and the discrete group \({\mathbb{H}}^1(K,M^*)\). Here the groups \({\mathbb{H}}^0(K,M)^{\wedge}\) and \({\mathbb{H}}^{-1}_{\wedge}(K,M)\) are obtained from the corresponding hypercohomology groups by certain completion procedures.
Let \(M\) be a \(1\)-motive over a number field \(k\), and for all \(i\geq 0\), define the Tate-Shafarevich groups \(\text Ш^i(M)=\ker[{\mathbb{H}}^i(k,M)\to \prod_{v} {\mathbb{H}}^i(\hat{k}_v, M)]\) where the product is taken over completions \(k_v\) at all places of \(k\).
Theorem 2: Let \(k\) be a number field and \(M\) a \(1\)-motive over \(k\). Then there exist canonical pairing \(\text Ш^i(M)\times\textШ^{2-i}(M^*)\to {\mathbb{Q}}/{\mathbb{Z}}\) for \(i=0,1\), which are non-degenerate modulo divisible subgroups.
The duality isomorphisms are established by first constructing pairings using étale cohomology, and then by comparing them to the Galois cohomological one. A twelve-term Poitou-Tate type exact sequence is established for a \(1\)-motive \(M_k\) over a number field \(k\) assuming the finiteness of \(\text Ш^1(A_k)\) and \(\text Ш^1(A_k^*)\) where \(A_k\) is the abelian variety corresponding to \(M_k\).
Editorial Remark: See also the corrigendum [D. Harari and T. Szamuely, J. Reine Angew. Math. 632, 233–236 (2009; Zbl 1172.14029)].

MSC:

14L15 Group schemes
11G09 Drinfel’d modules; higher-dimensional motives, etc.
12G05 Galois cohomology
14G20 Local ground fields in algebraic geometry

Citations:

Zbl 1172.14029
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