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On the Γ-Factors of Motives II Documenta Math. 69 On the ¡-Factors of Motives II Christopher Deninger1 Received: June 2, 2000 Revised: February 20, 2001 Communicated by Peter Schneider Abstract. Using an idea of C. Simpson we describe Serre's local ¡- factors in terms of a complex of sheaves on a simple dynamical system. This geometrizes our earlier construction of the ¡-factors. Relations of this approach with the "-factors are also studied. 1991 Mathematics Subject Classi¯cation: 11G40, 14Fxx, 14G10, 14G40 Keywords and Phrases: ¡-factors, motives, non-abelian Hodge theory, "-factors, explicit formulas, Rees bundle, dynamical systems 1 Introduction n n In [S] Serre de¯ned local Euler factors Lp(H (X); s) for the \motives" H (X) where X is a smooth projective variety over a number ¯eld k. The de¯nition at the ¯nite places p involves the Galois action on the l-adic cohomology groups n H¶et(X ­ kp; Ql). At the in¯nite places the local Euler factor is a product of Gamma factors determined by the real Hodge structure on the singular n cohomology HB(X ­ kp; R). If p is real then the Galois action induced by complex conjugation on kp has to be taken into account as well. Serre also conjectured a functional equation for the completed L-series, de¯ned as the product over all places of the local Euler factors. In his de¯nitions and conjectures Serre was guided by a small number of exam- ples and by the analogy with the case of varieties over function ¯elds which is quite well understood. Since then many more examples over number ¯elds no- tably from the theory of Shimura varieties have con¯rmed Serre's suggestions. The analogy between l-adic cohomology with its Galois action and singular cohomology with its Hodge structure is well established and the de¯nition of the local Euler factors ¯ts well into this philosophy. However in order to prove the functional equation in general, a deeper understanding than the one 1supported by TMR Arithmetic Algebraic Geometry Documenta Mathematica 6 (2001) 69{97 70 C. Deninger provided by an analogy is needed. First steps towards a uniform description of the local Euler factors were made in [D1], [D2], [D3]. There we constructed n in¯nite dimensional complex vector spaces Fp(H (X)) with a linear flow such that for all places: 1 ¡1 L (Hn(X); s) = det (s ¢ id ¡ £) j F (Hn(X)) : (1) p 1 2¼ p ³ ´ Here £ is the in¯nitesimal generator of the flow and det1 is the zeta-regularized determinant. Unfortunately the construction of the spaces n Fp(H (X)) was not really geometric. They were obtained by formal construc- tions from ¶etale cohomology with its Galois action and from singular cohomol- ogy with its Hodge structure. C. Consani [C] later developed a new in¯nite-dimensional cohomology theory n HCons (Y ) with operators N and £ for varieties Y over R or C such that for in¯nite places p: n » n N=0 (Fp(H (X)); £) = (HCons (X ­ kp) ; £) : (2) Her constructions are inspired by the theory of degenerations of Hodge struc- ture and her N has to be viewed as a monodromy operator. The formula for the archimedean local factors obtained by combining (1) and (2) is analogous to n the expression for Lp(H (X); s) at a prime p of semistable reduction in terms of log-crystalline cohomology. The conjectural approach to motivic L-functions outlined in [D7] suggests the n following: It should be possible to obtain the spaces Fp(H (X)) for archime- dian p together with their linear flow directly by some natural homological construction on a suitable non-linear dynamical system. Clearly, forming the intersection of the Hodge ¯ltration with its complex conjugate and running the resulting ¯ltration through a Rees module construction as in our ¯rst construc- n tion of Fp(H (X)) in [D1] is not yet what we want: In this construction the linear flow appears only a posteriori on cohomology but it is not induced from a flow on some underlying space by passing to cohomology. In the present paper in Theorems 4.2, 4.3, 4.4 we make a step towards this goal of a more direct dynamical description of the archimedian Gamma-factors. The approach is based on a result of Simpson which roughly speaking replaces the consideration of the Hodge ¯ltration by looking at a relative de Rham complex with a deformed di®erential. In our case, instead of the Hodge ¯ltration F ² we require the non-algebraic ¯ltration F ² \ F ². This forces us to work in a real analytic context even for complex p. It seems di±cult to carry Simpson's method over to this new context. However this is not necessary. By a small miracle { the splitting of a certain long exact sequence { his result can be brought to bear directly on our more complicated situation. In the appendix to section 4 we explain a relation between Simpson's deformed complex and a relative de Rham complex on the deformation of X to the normal Documenta Mathematica 6 (2001) 69{97 On the ¡-Factors of Motives II 71 bundle of a base point. This observation probably holds the key for a complete dynamical understanding of the Gamma-factor. Our main construction also provides a C!-vector bundle on R with a flow. Its ¯bre at zero can be used for a \dynamical" description of the contribution from p j 1 in the motivic \explicit formulas" of analytic number theory. In our investigation we encounter a torsion sheaf whose dimension is to some extent related to the "-factor at p of H n(X). Using forms with logarithmic singularities one can probably deal more generally with the motives Hn(X) where X is only smooth and quasiprojective. It would also be of interest to give a construction for Consani's cohomology theory using the methods of the present paper. I would like to thank J. Wildeshaus for discussions which led to the appendix of section 4. A substantial part of the work was done at the IUAV in Venice where I would like to thank U. Zannier and G. Troi very much for their hospitality. I would also like to thank the referee for a number of suggestions to clarify the exposition. 2 Preliminaries on the algebraic Rees sheaf In this section we recall and expand upon a simple construction which to any 1 1 ¯ltered complex vector space attaches a sheaf on A = AC with a Gm-action. We had used it in earlier work on the ¡-factors [D1], [D3] x 5. Later Simpson [Si] gave a more elegant treatment and proved some further properties. Most importantly for us he proved Theorem 5.1 below which was the starting point for the present paper. In the following we also extend his results to a variant of the construction where one starts from a ¯ltered vector space with an in- volution. This is necessary later to deal not only with the complex places but with the real places as well. Let FilC be the category of ¯nite dimensional complex vector spaces V with a descending ¯ltration FilrV such that Filr1 V = 0; Filr2 V = V for some integers § r1; r2. Let FilR be the category of ¯nite dimensional complex vector spaces with a ¯ltration as above and with an involution F1 which respects the ¯ltration. § Finally let FilR be the full subcategory of FilR consisting of objects where F1 induces multiplication by (¡1)² on Gr²V . These additive categories have ­-products and internal Hom's. We de¯ne Tate twists for every integer n by ² ²+n (V; Fil V )(n) = (V; Fil V ) in FilC and by ² ²+n n § (V; Fil V; F1)(n) = (V; Fil V; (¡1) F1) in FilR and FilR : Note that the full embedding: § i : FilR ,! FilR Documenta Mathematica 6 (2001) 69{97 72 C. Deninger is split by the functor § s : FilR ¡! FilR r which sends (V; Fil V; F1) to r r r+1 r r (¡1) (¡1) +1 (Vf; Fil V = (Fil V ) + (Fil V ) ; F1) §1 i.e. s ± i = id. Here W denotes the §1 eigenspace of F1 on W . For V in f f ² 1 FilC following [Si] x 5 de¯ne a locally free sheaf »C(V ) = »C(V; Fil V ) over A with action of Gm by p ¡p 1 »C(V ) = Fil V ­ z OA ½ V ­ j¤OGm : p X 1 1 Here j : Gm ,! A is the inclusion and z denotes a coordinate on A determined up to a scalar in C¤. Unless stated otherwise the constructions in this paper ² are independent of z. The global sections of the \Rees sheaf" »C(V; Fil V ) form the \Rees module" over C[z]: 0 ¡1 p ¡p ¡1 Fil (V ­C C[z; z ]) = Fil V ­ z C[z] ½ V ­ C[z; z ] p X p ¡1 p 1 where Fil C[z; z ] = z C[z] for p 2 Z. The natural action of Gm on A induces a Gm-action on »C by pullback ¤ ¡1 ¸ : (¸) »C ¡! »C ; v ­ g(z) 7¡! v ­ g(¸z) : (3) ¡1 Here (¸) »C denotes the inverse image of »C under the multiplication by ¸ 2 C¤ map. 1 1 2 1 1 Let sq : A ! A be the squaring map sq(z) = z and de¯ne F1 : A ! A as § 1 F1 = ¡id. For V in FilR the actions of F1 on V and Gm ½ A combine to an action ¤ ¡1 F1 : F1 (V ­ j¤OGm ) ¡! V ­ j¤OGm : Thus we get an involution F1 on the sheaf sq¤(V ­ j¤Gm) and we de¯ne a locally free sheaf on A1 by: ² ² F1 »R(V ) = »R(V; Fil V; F1) = (sq¤»C(V; Fil V )) : The Gm-action on »C leads to an action ¤ 2 ¡1 ¸ : (¸ ) »R ¡! »R : ² The global sections of »R(V; Fil V; F1) are given by F1 p ¡p ¡1 Fil V ­ z C[z] ½ V ­ C[z; z ] p ³ X ´ viewed as a C[z2]-module.
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