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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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References:
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