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Local $$L$$-factors of motives and regularized determinants. (English) Zbl 0762.14015
In a previous paper by the author [Invent. Math. 104, No. 2, 245–261 (1991; Zbl 0739.14010)] it was shown, among other things, that the local Euler factors of the $$L$$-function of a motive over a number field at the archimedean (infinite) places can be written as regularized determinants. Here the non-archimedean places are treated, thus giving a unified approach to all $$L$$-factors.
Let $$K$$ be a non-archimedean local field with inertia group $$I$$ and residue field $$\kappa\simeq\mathbb F_q$$, $$q=p^f$$ for some prime $$p$$. Write $$W_K$$ and $$W_K'$$ for the associated Weil group, resp. Weil-Deligne group, and $$\text{Rep}(W_K')$$ for the Tannakian category of finite dimensional complex representations of $$W_K'$$. The group ring $$\mathbb C [\mathbb C]$$ consists of elements of the form $$\sum re^\alpha$$, where $$r,\alpha\in\mathbb C$$ and the symbols $$e^\alpha$$ obey the rule $$e^{\alpha+\alpha'}=e^\alpha e^{\alpha'}$$. $$B$$ denotes the $$\mathbb C$$-algebra $$\mathbb C[\mathbb C]$$ equipped with two structures:
(i) an unramified representation of $$W_K'$$ such that $$\rho(w)(e^\alpha)=\| w\|^\alpha\cdot e^\alpha$$ for all $$w\in W_K$$, where $$\rho$$ defines the representation of $$W_K'$$;
(ii) a $$\mathbb C$$-linear derivation $$\Theta$$ defined by $$\Theta(e^\alpha)=\alpha e^\alpha$$.
Finally, let $$\mathbb L=B^{W_K'}$$ denote the $$\mathbb C$$-algebra of Laurent polynomials in $$e^{\alpha_ q}$$ with $$\alpha_q=2\pi i/\log q$$, and let $$\Delta=\mathbb L(\Theta)$$. The additive category of left $$\Delta$$-modules which are free of finite rank over $$\mathbb L$$ is written $$\text{Der}_ \kappa$$.
One defines an additive functor $$\mathbb D:\text{Rep}(W_K')\to\text{Der}_ \kappa$$ by $$\mathbb D(H)=(H\otimes B)^{W_K'}=(H_ N\otimes B)^{\rho(W_K)}$$ with $$H_ N$$ the kernel of the nilpotent endomorphism $$N$$ occurring in the definition of the representation $$H$$ of $$W_K'$$. In general one has $$\text{rk}_\mathbb L\mathbb D(H)\leq\dim H^I_N$$, and in case of equality, the representation $$H$$ is called admissible. One obtains a full semisimple Tannakian subcategory of admissible representations $$\text{Rep}^{\text{ad}}(W_K')$$.
On the other hand, one defines an additive functor $$\mathbb H: \text{Der}_\kappa\to\text{Rep}^{\text{ad}}(W_K')$$ by $$\mathbb H(D)=(D\otimes B)^{\Theta=0}$$ with $$W_K'$$-action induced by the one on $$B$$. In general $$\dim\mathbb H(D)\leq\text{rk}_\mathbb L D$$. $$D$$ is called admissible if there is equality. $$\text{Der}_ \kappa^{\text{ad}}$$ will denote the full subcategory in $$\text{Der}_\kappa$$ of admissible objects. Several characterizations of admissibility can be given. The first result now says:
The functors $$\mathbb D$$ and $$\mathbb H$$ provide quasi- inverse equivalence of the Tannakian categories $$\text{Rep}^{\text{ad}}(W_K')$$ and $$\text{Der}_\kappa^{\text{ad}}$$, commuting with tensor products, twists and duals.
As usual, for a representation $$H$$ in $$\text{Rep}(W_ K')$$, the local $$L$$-factor is given by
$L_K(H,s) = \det(1-\rho(\Phi)q^{-s}| H^I_N)^{-1},$
where $$\Phi$$ is the geometric Frobenius. For the maximal admissible submodule $$H^{\text{ad}}$$ of $$H$$, the following result can be shown:
$L_K(H^{\text{ad}},s) = \det_\infty\left(\frac{\log q}{2\pi i}(s-\Theta)|\mathbb D(H)\right)^{-1}$
where $$\det_\infty$$ denotes the regularized determinant. Writing $$\mathcal M_K$$ for the category of (pure) Deligne motives, i.e. motives for absolute Hodge cycles over $$K$$, and fixing an embedding $$\sigma: \mathbb Q_ \ell\hookrightarrow\mathbb C$$, one has the realization functor, believed to be independent of $$\ell$$ and $$\sigma$$, $$H_{\ell,\sigma}: \mathcal M_K\to\text{Rep}(W_ K')$$, $$\ell\neq p$$, given by $$H^\bullet_{\ell,\sigma}(M)=H^ \bullet_ \ell(M)\otimes_{\mathbb Q_ \ell,\sigma}\mathbb C$$. Denote by $$\mathcal M_K^{\text{ad}}$$ the subcategory of $$\mathcal M_K$$ with objects $$M$$ such that $$H^\bullet_{\ell,\sigma}(M)\in\text{Rep}^{\text{ad}}(W_ K')$$. For $$M$$ in $${\mathcal M}_K^{\text{ad}}$$ one sets $$H^\bullet(M/\mathbb L)=\mathbb D H^\bullet_{\ell,\sigma}(M)$$. For a smooth projective variety over $$K$$ such that the associated motive $$h(X)$$ is in $${\mathcal M}_K^{\text{ad}}$$, one writes $$H^\bullet(X/\mathbb L)$$ for $$H^\bullet(h(X)/\mathbb L)$$. Also, for any $$M$$ in $${\mathcal M}_K$$, one writes $$L_ K(H^w(M),s)=L_K(H^w_{\ell,\sigma}(M),s)$$. For $$M=h(X)$$ where $$X$$ has good reduction this local factor is known to be independent of $$\ell$$ and $$\sigma$$. The following theorem is proven: For $$X$$ smooth projective over $$K$$, and $$w=0,1$$ one has
$\det_ \infty\left(\frac{\log q}{2\pi i}(s-\Theta)|\mathbb D H^ w_{\ell,\sigma}(X)\right)^{-1}=L_ K(H^ w(X),s),$
in particular, if $$X$$ has good reduction the $$\mathbb D H^ w_{\ell,\sigma}(X)$$ may be replaced by $$H^ w(X/\mathbb L)$$. The theorem is expected to hold for all $$w$$.
The theorem suggests a global cohomological approach to $$L$$-functions of varieties over number fields. Concretely, there should exist a big site containing $$\overline{\text{Spec}(\mathbb Z)}=\text{Spec}(\mathbb Z)\cup\{\infty\}$$ with suitable properties. This formalism should give rise to some remarkable expressions involving the Riemann zeta-function. One result can indeed be proven by analytic means: For $$\operatorname{Re} z>1$$ consider the Dirichlet series $$\xi(s,z)=\sum_ \rho\frac{1}{\bigl[\frac{1}{2\pi}(z-\rho)\bigr]^ s}$$, where $$\rho$$ runs over the nontrivial zeros of the Riemann zeta-function and such that $$\arg(z-\rho)\in\bigl(-\frac{\pi}{2},\frac{\pi}{2}\bigr)$$. Then $$\xi(s,z)$$ converges absolutely for $$\operatorname{Re} s>1$$ and for fixed $$x$$ it admits an analytic continuation to a holomorphic function in $$\mathbb C\backslash\{1\}$$. One has the formula:
$2^{- 1/2}(2\pi)^{-2}\pi^{-z/2}\Gamma\left(\frac{z}{2}\right)\zeta(z)z(z-1)=\exp(-(\partial_ s\xi)(0,z)).$

##### MSC:
 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14A20 Generalizations (algebraic spaces, stacks) 11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) 11M38 Zeta and $$L$$-functions in characteristic $$p$$
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##### References:
 [1] [A] Ahlfors, L.V.: Complex Analysis. New York: McGraw-Hill 1966 · Zbl 0154.31904 [2] [Ba] Barner, K.: On A. Weil’s explicit formula. J. Reine Angew. Math.323, 139-152 (1981) · Zbl 0446.12013 [3] [C-V] Cartier, P., Voros, A.: Une nouvelle interpr?tation de la formule des traces de Selberg. (Grothendieck Festschrift II) Boston Basel Stuttgart: Birkh?user 1991 [4] [D] Deninger, C.: On the ?-factors attached to motives. Invent. Math.104, 245-261 (1991) · Zbl 0739.14010 [5] [E] Erdelyi, A. et al.: Higher transcendental functions, vol. I. Bateman Manuscript project. New York: McGraw-Hill 1953 [6] [F] Fontaine, J.-M.: Modules galoisiens, modules filtr?s et anneaux de Barsotti-Tate. In: Journ?e de G?om?trie Alg?brique de Rennes (Ast?risque, vol. 65, pp. 3-80) Paris: Soc. Math. de France 1979 [7] [G] Grothendieck, A.: Mod?les de N?ron et Monodromie. Exp IX in SGA 7, I. (Lect. Notes Math., vol. 288) Berlin Heidelberg New York: Springer 1972 [8] [J1] Jannsen, U.: On thel-adic cohomology of varieties over number fields and its Galois cohomology. In: Y. Ihara, K. Ribet, J.-P. Serre (eds.) Galois Groups over ?, pp. 315-353. (Publ., Math. Sci. Res. Inst., vol. 16) Berlin Heidelberg New York: Springer 1989 [9] [J2] Jannsen, U.: Motivic cohomology,l-adic cohomology and vanishing orders ofL-functions. (Preprint 1990) [10] [K1] Kurokawa, N.: Parabolic components of zeta functions. Proc. Japan Acad, Ser. A64, 21-24 (1988) · Zbl 0642.10028 [11] [K2] Kurokawa, N.: Analyticity of Dirichlet series over prime powers. (Lect. Notes Math., vol. 1434, pp. 168-177) Berlin Heidelberg New York: Springer 1990 [12] [K3] Kurokawa, N.: Multiple zeta functions: an example. In: Zeta functions in geometry. Adv. Stud. Pure Math. 1991 (to appear) [13] [Se] Serre, J.P.: Facteurs locaux des fonctions z?ta des vari?t?s alg?briques (d?finitions et conjectures). S?minaire Delange-Pisot-Poitou, expos? 19, 1969/70 [14] [So] Soul?, Ch.: Letter to the author. February 13, 1991 [15] [Ta] Tate, J.: Number theoretical background. In: A. Borel, W. Casselman (eds.): Automorphic forms, representations andL-functions, pp. 3-26. Corvallis 1977. (Proc. Symp. Pure Math. XXXIII, 2) Providence: Am. Math. Soc. 1979 [16] [V] Voros, A.: Spectral functions and the Selberg zeta function. Commun. Math. Phys.111, 439-465 (1987) · Zbl 0631.10025
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