A conjecture of A.J. de jong
Let \(X\) be an integral scheme, separated and of finite type over a finite field \(k\) of characteristic \(p\). Let \(\ell\) be a prime number different from \(p\). Assume that \(X\) is a normal scheme. Let \(\bar{\rho}:\pi_{1}(X)\to GL_{n}(\mathbb{F}_{\ell})\) be a continuous, absolutely irreducible representation of the arithmetic etale fundamental group of \(X\). Let \(\mathcal{O}\) be the ring of integers of a finite extension of the fraction field of \(\mathbb{Z}_{\ell}\) inside \(\overline{\mathbb{Q}_{\ell}}\). Fix a lift \(\eta:\pi_{1}(X)\to \mathcal{O}^{\times}\) of finite order of the \(1\)-dimensional representation \(\text{det}(\bar{\rho})\). Then we get a deformation ring \(R_{X}^{\eta}\) for deformations of \(\bar{\rho}\) of determinant \(\eta\) and defined on \(\pi_{1}(X)\).
In the situation above, assume
- the restriction \(\bar{\rho}\vert_{\pi_{1}(\overline{X})}\) is absolutely irreducible, and
- \(\ell\) does not divide \(n\).
Then \(R_{X}^{\eta}\) is finite flat over \(\mathbb{Z}_{\ell}\) and it is a complete intersection ring. In particular, \(\bar{\rho}\) can be lifted to an \(\ell\)-adic representation \(\rho:\pi_{1}(X)\to GL_{n}(\mathcal{O})\).
A question
Let \(X\) be a scheme over \(\mathbb{F}_{p}\). Let \(\ell\) be a prime which is not equal to \(p\). Let \(\rho:\pi_{1}(X)\to GL_{d}(\overline{\mathbb{F}_{\ell}})\) be a representation of the arithmetic 'etale fundamental group of \(X\). We consider ‘the characteristic polynomials map’:
\[P_{\rho}:\pi_{1}(X)\to \overline{\mathbb{F}_{\ell}}[X],~~g\mapsto P_{g}:=\text{characteristic polynomail of $ \rho(g) $}.\]There exists an integer \(k\) such that all \(P_{g}\) split over \(\mathbb{F}_{\ell^{k}}\). For any \(g\in G\), if \(P_{g}=\prod_{i=1}^{d}(X-\alpha_{i})\) for some \(\alpha_{i}\in \mathbb{F}_{\ell^{d}}\), then we define
\[\tilde{P}_{g}:=\prod_{i=1}^{d}(X-\tau(\alpha_{i}))\in W(\mathbb{F}_{\ell^{k}})[X]\]where \(\tau: \mathbb{F}_{\ell^{d}}\to W(\mathbb{F}_{\ell^{d}})\) is the Teichmuller representation. Fix an isomorphism of the algebraic closure of \(\mathbb{Q}\) within \(\overline{\mathbb{Q}_{\ell}}\) and \(\overline{\mathbb{Q}_{p}}\). We may regard \(\tilde{P}_{g}\) as an element in \(\overline{\mathbb{Z}_{p}}[X]\). Then we define
\[Q_{g}:=\text{the image of $ \tilde{P}_{g} $ in $ (\overline{\mathbb{Z}_{p}}/p\overline{\mathbb{Z}_{p}})[X] $}\]Question:
Is there a representation \(\rho':\pi_{1}(X)\to GL_{d}(\overline{\mathbb{F}_{p}})\) such that, for any \(g\in \pi_{1}(X)\), the characteristic polynomial of \(\rho'(g)\) is \(Q_{g}\)?