Arithmetic properties of local systems on algebraic varieties

Abstract

The first main result of this dissertation is a statement about the ubiquity of de Rham local systems on algebraic varieties over $p$-adic fields. If $\mathbb{L}$ is a geometrically irreducible $\overline{\mathbb{Q}}_p$-local system on a smooth algebraic variety over a finite extension $K$ of $\overline{\mathbb{Q}}_p$ then there exists a character of the Galois group $\chi:G_K\to \overline{\mathbb{Q}}_p^{\times}$ such that the twist $\mathbb{L}\otimes\chi$ is a de Rham local system. The proof relies on a decompleted version of the $p$-adic Riemann-Hilbert correspondence of Liu and Zhu. We also prove a version of this result for local systems on $X$ with arbitrary geometric monodromy. That generalization can be restated as saying that the Galois action on the pro-algebraic completion of the 'etale fundamental group $\pi_1^{\et}(X_{\overline{K}})$ is de Rham. In particular, if $S$ is a smooth variety over a number field $F$ then every Galois representation appearing as a subquotient of the space of algebraic functions on $\pi_1^{\et}(S_{\overline{F}})$ is geometric in the sense of Fontaine-Mazur. Our second main result is a universality statement regarding the Galois action on the space of algebraic functions on the 'etale fundamental group. We prove that any semi-simple representation of the Galois group of $F$ that arises from geometry can be established as a subquotient of the space of regular functions on $\pi_1^{\et}(\mathbb{P}^1_{\overline{F}}\setminus{0,1,\infty})$. Combining the above-mentioned results implies, assuming the truth of the Fontaine-Mazur conjecture, that the class of semi-simple subquotients of this space of functions is precisely the class of all semi-simple Galois representations of geometric origin.