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Motivic \(L\)-functions and regularized determinants. II. (English) Zbl 1163.11338

Catanese, Fabrizio (ed.), Arithmetic geometry. Proceedings of a symposium, Cortona, Arezzo, Italy, October 16–21, 1994. Cambridge: Cambridge University Press (ISBN 0-521-59133-3/hbk). Symp. Math. 37, 138-156 (1997).
Introduction: In the papers [[De1] P. Deligne, Equations différentielles à points singuliers réguliers, Lect. Notes Math. 163, Berlin etc.: Springer (1970; Zbl 0244.14004); [De2] Publ. Math., Inst. Hautes Étud. Sci. 40, 5–57 (1971; Zbl 0219.14007)] we gave an interpretation of local \(L\)-factors of pure motives as regularized characteristic power series on infinite dimensional cohomologies. This lead to speculation on an “arithmetic site” whose global cohomologies would be deeply connected with the global \(L\)-series of motives. These arguments suggested in particular a formula for the Riemann \(\xi\)-function as a regularized characteristic power series which was proved in [De2, §4].
In sections 1 to 6 of this article we extend the above interpretation to the local \(L\)-factors of mixed motives. For the finite primes we give an improved construction of the infinite dimensional cohomologies using an elementary case of the Riemann-Hilbert correspondence. This does away with the semisimplicity assumption we had to make in [De2]. This new point of view was noted independently by S. Bloch. We also understand better than in [De1] the relation between Archimedian and Deligne cohomology.
Apart from this our main objective is to discuss in some detail the following aspects of the still speculative “arithmetic cohomology”:
What form should a Lefschetz fixed point formula take? We mention the relation with explicit formulas in analytic number theory.
We give a short “proof” in the spirit of [J.-P. Serre, Ann. Math. (2) 71, 392–394 (1960; Zbl 0203.53601)] of the Riemann hypotheses assuming that a Hodge \(*\)-operator with standard properties exists on the prospected cohomologies.
Following a classical pattern we relate the functional equation for motivic \(L\)-series to Poincaré duality.
We “explain” the well-known conjectures on the vanishing and pole order of \(L\)-functions at integers by certain cohomological conjectures. We point out relations between a Künneth formula and Kurokawa’s multiple zeta functions.
In sections 1 to 6 everything is proved and we think of mixed motives in terms of realizations [P. Deligne, Galois groups over \(\mathbb Q\), MSRI Publications, Springer, 79–297 (1989; Zbl 0742.14022), U. Jannsen, Mixed motives and algebraic K-theory. Lect. Notes Math. 1400. Berlin etc.: Springer (1990; Zbl 0691.14001)]. In the speculative §7 we are not precise about the meaning of the word motive in the formal discussions.
For the entire collection see [Zbl 0864.00054].

MSC:

11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
14A20 Generalizations (algebraic spaces, stacks)
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