From the text: Let \(X\) be a separated finite type \(\mathbb{F}_p\)-scheme and \({\mathcal L}\) a lisse \(\mathbb{Z}_p\)-sheaf on the étale site of \(X\). The \(L\)-function \(L(X,{\mathcal L})\) attached to \({\mathcal L}\) is a power series in a formal variable \(T\), which by construction is an element of \(1+T\mathbb{Z}_p [[T]]\). If \(f:X\to\text{Spec}\,\mathbb{F}_p\) is the structural morphism of \(X\) then \(f_!{\mathcal L}\) is a constructible complex of \(\mathbb{Z}_p\)-sheaves on the étale site of \(\text{Spec}\, \mathbb{F}_p\) of finite tor-dimension, and so we may form the \(L\)-function \(L(\text{Spec}\,\mathbb{F}_p\), \(f_!{\mathcal L})\), which is also an element of \(1+T\mathbb{Z}_p[[T]]\) (and in fact a rational function). The ratio \(L(X,{\mathcal L})/L(\text{Spec}\, \mathbb{F}_p,f_!{\mathcal L})\) thus lies in \(1+T\mathbb{Z}_p[[T]]\), and so may be regarded as a nowhere-zero analytic function on the \(p\)-adic open unit disk \(|T|<1\). If \({\mathcal L}\) were a lisse \(\mathbb{Z}_l\) sheaf with \(l\neq p\) then in fact this ratio would be identically 1 (this is Grothendieck’s approach to the rationality of the zeta function). One does not have this in the setting of \(\mathbb{Z}_p\)-sheaves. However, in this paper we prove the following result, which was conjectured by N. Katz [in: Sem. Bourbaki 1971/72, No. 409, Lect. Notes Math. 317, 167–200 (1973; Zbl 0259.14007)].
Theorem 1. The ratio \(L(X,{\mathcal L})/L(\text{Spec}\,\mathbb{F}_p,f_!{\mathcal L})\) extends to a nowhere-zero function on the closed unit disc \(|T|\geq 1\). In particular this implies that the \(L\)-function for \(L(X,{\mathcal L})\) is \(p\)-adic meromorphic in the closed unit disc. Katz also conjectured that \(L(X,{\mathcal L})\) extends to a meromorphic function on the rigid-analytic affine \(T\)-line. The second part of Katz’s conjecture was a generalization of a conjecture of Dwork predicting that \(L(X,{\mathcal L})\) was \(p\)-adic meromorphic when the unit \(F\)-crystal \({\mathcal M}\) associated to \({\mathcal L}\) was the unit part of an ordinary, overconvergent \(F\)-crystal. D. Wan [Ann. Math. (2) 143, No. 3, 469–498 (1996; Zbl 0868.14011)] has found counter-examples to the part of Katz’s conjecture predicting that the \(L\)-function is meromorphic for general lisse sheaves. On the other hand, in a recent preprint he has proven that Dwork’s more cautious prediction is actually true. Thus, while the question of when the \(L\)-function is meromorphic is by now quite well understood, much less was known about the location of the zeroes and poles. In fact we prove a more general result showing that an analogue of Katz’s conjecture is true even if one replaces \(\mathbb{Z}_p\) by a complete noetherian local \(\mathbb{Z}_p\)-algebra \(\Lambda\) with finite residue field. Our methods also have some applications to results on lifting representations of arithmetic fundamental groups. In particular, we show the following
Theorem 2. Let \(X\) be a smooth affine \(\mathbb{F}_p\)-scheme, \(\Lambda\) an artinian local \(\mathbb{Z}_p\)-algebra, having finite residue field, and \[ \rho:\pi_1(X)\to\text{GL}_d (\Lambda) \] a representation of the arithmetic étale fundamental group of \(X\). Consider a finite flat local \(\mathbb{Z}_p\)-algebra \(\widetilde\Lambda\) and a surjection \(\widetilde\Lambda\to\Lambda\). There exists a continuous lifting of \(\rho\) \[ \widetilde\rho:\pi_1(X)\to\text{GL}_d (\widetilde\Lambda). \] If \(X\) is an open affine subset of \(\mathbb{P}^1\), then \(\widetilde\rho\) may be chosen so that the \(L\)-function of the corresponding lisse sheaf of \(\widetilde\Lambda\)-modules is rational.