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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).
[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)

##### MSC:
 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