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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.

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
Full Text: DOI EuDML
[1] [B] Beilinson, A.A.: Higher regulators and values ofL-functions. J. Sov. Math.30, 2036-2070 (1985) · Zbl 0588.14013
[2] [E] Erdelyi, A.: Higher transcendental functions Vol. I. Bateman Manuscript project. New York Toronto London: McGraw-Hill 1953
[3] [F1] Fontaine, J.-M.: Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate. In: Journée dé Géométrie Algébrique de Rennes, Astérisque65, 3-80. Société Math. de France, Paris 1979
[4] [F2] Fontaine, J.-M.: Sur certains types de représentationsp-adique du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate. Ann. Math.115, 529-577 (1982) · Zbl 0544.14016
[5] [H] Hawking, S.W.: Zeta function regularization of Path Integrals in curved spacetime. Comm. Math. Phys.55, 133-148 (1977) · Zbl 0407.58024
[6] [J] Jannsen, U.: Deligne homology, Hodge-D-conjectures and motives. In: Rapoport, M., Schappacher, N., Schneider, P. (eds.) Beilinson’s conjectures on special values ofL-functions, 305-372, (Perspectives in Math., Vol. 4) Boston New York: Academic Press 1988
[7] [R-S] Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds. Ann. Math.98, 154-177 (1973) · Zbl 0267.32014
[8] [Sch] Schneider, P.: Introduction to the Beilinson conjectures. In: Rapoport, M., Schappacher, N., Schneider, P. (eds) Beilinson’s conjectures on special values ofL-functions, 1-35. (Perspectives in Math., Vol. 4) Boston New York: Academic Press 1988
[9] [Se1] Serre, J.P.: Sur les groupes de Galois attachés aux groupesp-divisibles. In: Proceedings of a conference on local fields, Nuffic Summer school at Driebergen, 118-131, Berlin Heidelberg New York: Springer 1967
[10] [Se2] Serre, J.P.: Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures). Sém. Delange-Pisot-Poitou, exp. 19, 1969/70
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