## 1-motives and geometric monodromy. (Exposé VII: 1-motifs et monodromie géométrique.)(French)Zbl 0830.14001

Fontaine, Jean-Marc (ed.), Périodes $$p$$-adiques. Séminaire de Bures- sur-Yvette, France, 1988. Paris: Société Mathématique de France, Astérisque. 223, 295-319 (1994).
For a scheme $$S$$ an $$S$$-$$1$$-motive $$M = [Y \to G]$$ is herein defined by
(i) an $$S$$-group scheme $$Y$$ locally (in the étale topology of $$S)$$ isomorphic to $$\mathbb{Z}^r$$,
(ii) a commutative $$S$$-group scheme $$G$$, extension of an abelian scheme $$A$$ by a torus $$T$$, and
(iii) an $$S$$-homomorphism $$u : Y \to G$$.
If $$S$$ is the spectrum of an algebraically closed field the definition reduces to that given by P. Deligne [Inst. Hautes Étud. Sci., Publ. Math. 44 (1974), 5-77 (1975; Zbl 0237.14003)]. For $$S = \text{Spec} (R)$$ where $$R$$ is a complete discrete valuation ring and $$K$$ its function field, a $$K$$-$$1$$-motive with good reduction (resp. potentially good reduction) is defined by the property of yielding an $$R$$-$$1$$-motive (resp. after a finite extension of $$K)$$. To any $$K$$-$$1$$-motive $$M = [Y \to G]$$ is canonically associated a strict $$K$$-$$1$$-motive, i.e. $$M' = [Y' \to G']$$ such that $$G'$$ has potentially good reduction, a quasi- isomorphism $$M_{\text{rig}}' \to M_{\text{rig}}$$ in the derived category of bounded complexes of $$fppf$$-sheaves on the small rigid site of $$\text{Spec} (K)$$, producing a canonical isomorphism $$T_\ell (M') \cong T_\ell (M)$$ between the $$\ell$$-adic realizations, for any prime $$\ell$$. For a strict $$K$$-$$1$$-motive $$M = [Y \to G]$$, the geometric monodromy $$\mu : Y \times Y^* \to \mathbb{Q}$$ (where $$Y^*$$ is the character group of the torus $$T)$$ is defined by dealing with the trivialization of the Poincaré biextension, canonically associated with the Cartier dual of $$M$$. The geometric monodromy is zero if and only if $$M$$ has potentially good reduction. Finally, the geometric monodromy is related with the $$\ell$$-adic realization.
For the entire collection see [Zbl 0802.00019].

### MSC:

 14A20 Generalizations (algebraic spaces, stacks) 14G20 Local ground fields in algebraic geometry 14F30 $$p$$-adic cohomology, crystalline cohomology

### Citations:

Zbl 0237.14003; Zbl 0292.14005