Defining arithmetic local systems

Question

In his ICM 2018 lecture Venkatesh gives a "working" definition of arithmetic local systems of $K$-vector spaces for various fields $K$ as follows: [ICM 2018 Proceedings, Vol. 1, p. 278]

While for $\mathbb{F}_p$ and $\mathbb{Q}_p$ there is no ambiguity in the definitions, for $\mathbb{Q}$ it is not easy to see over which field to construct representations of arithmetic étale fundamental groups because $GL_n(\mathbb{Q})$ has "too few" finite subgroups, so tentatively he proposes Chow motives, but adds (on p. 277) that with Ayoub and Cisinski-Déglise's theories of motivic sheaves now available it might be possible to give a better definition. For $\mathbb{C}$ he says he has no idea and deems it a fundamental problem of number theory.

Q. 1: For $\mathbb{Q}$ what would be the benefits of using motivic sheaves in place of Chow motives? Would they furnish more local systems as needed for Langlands correspondence, or just be easier to work with? Has his suggestion actually been worked out using any of several constructions of category of motivic sheaves?

Q. 2: For $\mathbb{C}$ is there no definition because there is as yet no motivic Grothendieck group whose representations these local systems would arise from? Or is it some other reason? Reading the referenced paper of Arthur hasn't clarified it for me even though the paper is as clear as it could be.

Answer 1

For Q1: the problem is that Chow motives are a little too restrictive, because they are all (conjecturally at least) semi-simple, whereas non-semisimple objects are hugely important for arithmetic. E.g. if you take an elliptic curve $E / \mathbf{Q}$, and you let $Y = E - P - \infty$ where $P$ is a (non-trivial) rational point of $E$, then the excision sequence in etale cohomology shows that the cohomology of $Y$ contains a Galois representation which is an extension of $V_p(E)$ by the trivial representation in the category of $\mathbf{Q}_p$-linear Galois reps; and this extension realises the cohomology class attached to $P$ by Kummer theory. However, this "punctured elliptic curve" isn't a proper variety so it doesn't give a Chow motive. The various approaches to "mixed motives", including Ayoub and Cisinski--Deglise's works, are an attempt to construct a bigger category which includes Chow motives but also has some interesting non-semisimple objects, and which can "see" non-proper or non-smooth varieties.

For Q2: perhaps what Venkatesh is getting at here is the fact that there are lots of automorphic objects arising in the Langlands program, e.g. non-algebraic Maass forms for $GL(2)$, which are clearly number-theoretically important but are not "definable over $\overline{\mathbf{Q}}$" in any natural way -- e.g. they give $L$-functions $\sum a_n n^{-s}$ which have analytic continuation and functional equations, but such that the $a_n$'s are complex numbers which are (as far as we can tell) transcendental. These should correspond to some kind of "local system" which we don't have any idea as yet how to define.