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On the $$\Gamma$$-factors attached to motives. (English) Zbl 0739.14010
Let $$X$$ be a smooth projective variety over a number field $$k$$. Of prime interest to number theorists are the $$L$$-functions of the cohomology groups $$H^ i$$ for $$0\leq i\leq 2\dim(X)$$. This $$L$$-function is defined by an Euler product over all the primes of $$k$$. For finite primes one uses (essentially) the characteristic polynomial of the action of the Frobenius elements on the inertia-fixed part of the cohomology (in the manner of Artin $$L$$-series); for infinite primes one uses certain $$\Gamma$$-factors – see J.-P. Serre in Sémin. Théorie Nombres, Sémin. Delange-Pisot-Poitou 11 (1969/70; Zbl 0214.48403). The purpose of the paper under review is to put the finite and infinite primes on a more equal conceptual setting. Thus the author uses an archimedean analog of Fontaine’s ring $$B_{DR}$$ to define a cohomology theory $$H^*_{ar}$$ for smooth projective varieties over $$\mathbb R$$ or $$\mathbb C$$. These spaces are infinite dimensional and carry a natural endomorphism $$\Theta$$. The author then shows that at an infinite prime $$\nu$$ the $$\Gamma$$-factor can be obtained as $$(2\pi)^{-1}\det(s-\Theta\mid H^ i_{ar}(X_ \nu))^{-1}$$. Relations to Deligne cohomology are also given.

##### MSC:
 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11G35 Varieties over global fields 14F30 $$p$$-adic cohomology, crystalline cohomology 14G25 Global ground fields in algebraic geometry
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