Motivic \(\varepsilon\)-factors at infinity and regularized dimensions
Using results of other work of the author [cf. Motives, Proc. Summer Res. Conf. Motives, Univ. Washington 1991, Proc. Symp. Pure Math. 55, Part I, 707-743 (1994; Zbl 0816.14010)] one can rewrite the non-archimedean \(\varepsilon\)-factors of the \(L\)-function of a motive in terms of regularized superdimensions. The formula thus obtained suggests a similar result for the archimedean \(\varepsilon\)-factors. To obtain such a formula consider the categories \({\mathcal M} {\mathcal H}_ \mathbb{C}\) and \({\mathcal M} {\mathcal H}_ \mathbb{R}\) of real mixed Hodge structures and of real mixed Hodge structures with an involution \(F_ \infty\) respecting the weight filtration and such that its \(\mathbb{C}\)-linear extension maps \(F^ \bullet H_ \mathbb{C}\) into \(\overline F^ \bullet H_ \mathbb{C}\) for an object \(H\) of \({\mathcal M} {\mathcal H}_ \mathbb{R}\), respectively. Also, for \(K = \mathbb{R}\) or \(\mathbb{C}\) and a number field \(E\), write \({\mathcal M} {\mathcal H}_ K (E)\) for the \(E\)-linear category of objects of \({\mathcal M} {\mathcal H}_ K\) with multiplication by \(E\). Then on \({\mathcal M} {\mathcal H}_ K (E)\) one defines an \(E \otimes \mathbb{C}\)-invariant filtration \(\gamma^ \nu H_ \mathbb{C} = F^ \nu H_ \mathbb{C} \cap \overline F^ \nu H_ \mathbb{C}\). This filtration is used (by Fontaine and Perrin-Riou) to define the local \(\varepsilon\)- factors at infinity. On the other hand, let \({\mathcal F} il_ K\) denote the exact additive category of finite dimensional complex vector spaces with finite decreasing filtration and, in case \(K = \mathbb{R}\), there should exist an involution \(F_ \infty\) respecting this filtration and inducing \((- 1)^ \bullet\) on \(\text{Gr}^ \bullet V\) for \(V \in {\mathcal F} il_ \mathbb{R}\). In his paper cited above, the author defined functors \(\mathbb{D}\) (the Riemann-Hilbert correspondence) for the finite places and \(\mathbb{D}^ +\) from \({\mathcal F} il_ K\) to the category of \(\mathbb{L}_ K^ + [\Theta]\)- modules. The functor \(\mathbb{D}\) plays the key role in the construction of the non-archimedean local \(\varepsilon\)-factors. \(\mathbb{D}^ +\) is its natural archimedean counterpart. One defines two functors \({\mathcal V}, \text{Gr}^ \bullet_ F : {\mathcal M} {\mathcal H}_ K \to {\mathcal F} il_ K\). A formula analogous to the ones for the non-archimedean local \(\varepsilon\)-factors is derived for the archimedeans where the role of \(\mathbb{D}\) is played by the \(\mathbb{D}^ + ({\mathcal V} (H))\) and \(\mathbb{D}^ + (\text{Gr}^ \bullet_ F H_ \mathbb{C})\).
\(\varepsilon\)-factors of the \(L\)-function of a motive
scientific article; zbMATH DE number 749790