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4. Predictions of the Relative Fontaine-Mazur and Simpson conjectures

The relative Fontaine-Mazur conjecture predicts that arithmetic local systems are motivic, or “arise from geometry” i.e. they appear in the cohomology of families of smooth projective varieties. In particular, isolated points of the character variety are automatically arithmetic (since each Frobenius permutes the finitely many rigid points hence is trivial after some uniform power) predicting a conjecture of Simpson that rigid local systems arise from geometry.

Simpson also made further conjectures using the distict scheme structures on the three moduli spaces involved in the non-abelian Hodge correspondence $M_B$, $M_{\mathrm{dR}}$, and $M_{\mathrm{Dol}}$. If $X / \overline{\mathbb{Q}}$ is a variety over a number field, these moduli spaces each are canonically endowed with the structure of a $\overline{\mathbb{Q}}$-scheme but in noncompatible ways under the Riemann-Hilbert and Simpson correspondences. Furthermore, $M_{\mathrm{Dol}}$ is endowed with an algebrac $\mathbb{G}_m$-action. Simpson predicted that points – or more generally subvarieties – compatible with multiple algebraic and/or group-action structures are motivic. He conjectured:

(1) a $\mathbb{Z}$-local system underlying a $\mathbb{C}$-VHS (equivalently its corresponding Higgs bundle is $\mathbb{G}_m$-fixed and hence defined over $\overline{\mathbb{Q}}$) is motivic

(2) a $\mathbb{Z}$-local system whose associated flat bundle is defined over $\mathbb{Q}$ is motivic.

These conjectures are known in rank $1$ and partially in rank $2$ due to the work of Corlette–Simpson.


    1. Add remark

      Problem 4.1.

      [Yohan Brunebarbe] Suppose a flat bundle $(\mathcal{E}, \nabla)$ on a smooth projective variety $X / \overline{\mathbb{Q}}$ is defined over $\overline{\mathbb{\Q}}$ and its associated Higgs bundle is $\mathbb{G}_m$-fixed (equivalently the local system $\mathcal{E}^{\nabla}$ underlies a $\mathbb{C}$-VHS) then is it motivic?
          In rank $r = 1$ this is asking if a unitary flat bundle is defined over $\overline{\mathbb{Q}}$ does it have finite monodromy. The answer is yes, due to the work of [CITE].

        • Decidability of arithmeticity and motivicty


          Add remark

          Problem 4.2.

          Let $X / \mathbb{C}$ be a smooth projective variety. Is there an algorithm which given an integral point of the character variety $x \in M_B(X, r)(\mathcal{O}_K)$, where $K$ is a number field, can decide if $x$ is (after some embedding $\mathcal{O}_K \subset \mathbb{Q}_p$) an arithmetic local system? Likewise, is there an algorithm to determine if $x$ arises from geometry?
              Listing all families of smooth projective varieties, one can decide “yes” to $x$ being motivic so motivicity is recursively enumerable. By the work of Litt [MR4278666] one can decide “no” to $x$ being arithmetic so arithmeticity is co-recursively enumerable. Therefore, assuming the relative Fontaine-Mazur conjecture, the answer to both questions is “yes”.

            •     For a smooth projective variety over a finite field $X / \mathbb{F}_q$ we can ask the same question as before. This is interesting because in this case the relative Fontaine-Mazur conjecture is known by the work of Drinfeld and Lafforgue. However, it becomes more difficult to specify a representation without acess to the topological fundamental group.

              Add remark

              Problem 4.3.

              Let $X / \mathbb{F}_q$ be a smooth projective variety over a finite field. Given a representation $\rho : \pi_1^{\mathrm{et}}(X_{\overline{\mathbb{F}_q}}, \bar{x}) \to \mathrm{GL}_r(\mathbb{Z}_{\ell})$ (perhaps specified via an oracle returning its action on $(\mathbb{Z} / \ell^n \mathbb{Z})^r$ for each $n$ which is itself equivalent to the data of a certain finite etale cover and hence specified by a finite amount of data) is there an algorithm which decides if $\rho$ is arithmetic (equivalently motivic).
                  Again using the work of Litt, we can decide “no”. Furthermore, one can compute a trace function by point counting and hence deduce which arithmetic representation it must equal if it were arithemtic. However, without an a priori bound on the first power of $\ell$ where it could become different from an arithmetic representation, one cannot verify ”yes”.

                • Isolated singularities of the character variety

                      Similar to Simpson’s conjectures about rigid local systems, other “canonical” finite subsets of the character variety should also enjoy distinguished properties under the nonabelian Hodge correspondence. These include isolated singular points and isolated points of cohomology jump loci. We can also about all the corollaries of motivicity as is done in Simpson’s integrality conjecture for rigid local systems or other predictions such as the existence of an $F$-isocrystal structure on the reduction mod $p$.

                  Add remark

                  Problem 4.4.

                  Let $X / \mathbb{C}$ be a smooth quasi-projective variety and $x \in M_B(X, r)$ an isolated singularity or isolated point of a cohomology jump loci. Is the corresponding local system motivic? Can we prove any consequences of motivicity for this local system?
                      Nonabelian Hodge theory shows that the corresponding local system underlies a $\mathbb{C}$-VHS. Hence Simpson’s conjecture predicts these points are motivic. For almost all $p$, the methods of Esnault-Groechenig can be adapted to show that the corresponding flat bundle (mod $p$) has vanishing $p$-curvature and, in fact, underlies an $F$-isocrystal.

                      Cite this as: AimPL: Motives and mapping class groups, available at http://aimpl.org/motivesandmcg.