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:
\[ 2^{- 1/2}(2\pi)^{-2}\pi^{-z/2}\Gamma\left(\frac{z}{2}\right)\zeta(z)z(z-1)=\exp(-(\partial_ s\xi)(0,z)). \]