Duality and Polarization Form for Abelian Anderson T-motives

We introduce the notion of duality for an abelian Anderson T-motive M . Main results of the paper (all M have N = 0): 1. Algebraic duality implies analytic duality (Theorem 4.4). Explicitly, this means that a Siegel matrix of the dual of a uniformizable M is the minus transposed of a Siegel matrix of M . 2. There is a 1 – 1 correspondence between pure T-motives of dimension r − 1 and rank r (all they are uniformizable), and lattices of rank r in C ∞ having dual (Corollary 4.9.1.4). 3. Pure T-motives have duals which are pure T-motives as well (Theorem 2.3). 4. For a self-dual uniformizable M a polarization form on its lattice L(M) is defined. For some M this form is skew symmetric like in the number field case, but for some other M it is symmetric. An example is given. 5. We define Hodge filtration of cohomology and for the above self-dual M we formulate Hodge conjecture in codimension 1 (Section 4.8). 6. Some explicit results are proved for M having complete multiplication. The CM-type of the dual of M is the complement of the CM-type of M . Moreover, for M having multiplication by a division algebra there exists a simple formula for the CM-type of the dual of M (Section 5). 7. We construct a class of non-pure T-motives (T-motives having the completely non-pure row echelon form) for which duals are explicitly calculated (Theorem 3.5). This is the first step of the problem of description of all abelian T-motives having duals.