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\(p\)-adic Betti lattices. (English) Zbl 0739.14009
p-adic analysis, Proc. Int. Conf., Trento/Italy 1989, Lect. Notes Math. 1454, 23-63 (1990).

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[For the entire collection see Zbl 0707.00010.]
Let \(\overline\mathbb{Q}\) denote the algebraic closure of \(\mathbb{Q}\) in some fixed algebraic closure of \(\mathbb{Q}_ p\), say \(\overline\mathbb{Q}_ p\). Let \(A\) be an abelian variety over \(\overline\mathbb{Q}_ p\). According to the work of Fontaine and Messing, there is a canonical pairing between the Tate module \(T_ p(A_{\overline\mathbb{Q}_ p})\) and the de Rham cohomology \(H^ 1_{DR}(A)\) with values in a certain \(\overline\mathbb{Q}_ p\)-algebra \(\mathbb{B}_{DR}\). If \(A\) is defined over \(\overline\mathbb{Q}\), there is a natural \(\overline\mathbb{Q}\)-structure on \(H^ 1_{DR}\) and any embedding \(\gamma:\overline\mathbb{Q}\hookrightarrow\mathbb{C}\) gives a \(\mathbb{Z}\)- lattice \(H^ 1_ B(A_ \mathbb{C},\mathbb{Z})\) on \(T_ p(A_{\overline\mathbb{Q}_ p})\). By analogy with a conjecture of Grothendieck in the complex case, Fontaine asked whether the \(\overline\mathbb{Q}\)-transcendence degree of the values of this pairing restricted to these substructures is the dimension of the Mumford-Tate group. We disprove this, but show (under some mild hypothesis) that it is true for sufficiently general \(\gamma\).
For \(A/\overline\mathbb{Q}_ p\) of dimension \(g\) with multiplicative reduction, we then construct a \(p\)-adic Betti lattice of another kind. Using Frobenius one constructs a pairing between this rank \(2g\) \(\mathbb{Z}\)- lattice \({\mathcal L}\) and \(H^ 1_{DR}\) with values in a Laurent polynomial ring \(\overline\mathbb{Q}_ p[t,t^{-1}]\), depending on the choice of a branch of \(\log_ p\). In the relative case \({\mathcal L}\) is horizontal and the formula giving the pairing is quite similar to the complex one (\(p\)-adic evaluation of the same power series, \(\log\mapsto\log_ p\), \(2i\pi\mapsto t)\). We then study the \(p\)-adic analogue of the Grothendieck conjecture in this context, and suggest the possibility of a purely \(p\)-adic definition of Hodge cycles.
Recently all this was widely generalized in the construction of a \(\overline\mathbb{Q}\)-structure in the de Rham cohomology of any abelian variety or (conditionally) any projective variety with good reduction over \(\overline\mathbb{Q}_ p\), stable under correspondences and with horizontal behaviour in the relative situation.
Reviewer: Y.André (Paris)

14F30 \(p\)-adic cohomology, crystalline cohomology
14F40 de Rham cohomology and algebraic geometry
14K15 Arithmetic ground fields for abelian varieties
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
11S99 Algebraic number theory: local fields