First, as a disclaimer, I should say that this post is not about any specific propositions, but is more of some philosophical flavor.

In the world of quantum mathematics, the letter $q$ is a standard symbol that parametrizes deformation/quantization of a theory. Examples include (affine, double affine,...) Hecke algebra, (elliptic) quantum group.

The letter $q$ appears in a (seemingly) completely different way in arithmetic. Namely, it appears as the number of elements in the finite field $\mathbb{F}_q$, or fancier, eigenvalues of Frobenius automorphisms. In some cases, the formal limit $q\to 1,\mathbb{F}_q\to\mathbb{F}_1$ can be thought as some semiclassical limit.

There is an example when quantum $q$ equals to arithmetic $q$, that is, the famous Kazhdan--Lusztig conjecture. The conjecture (now theorem) indicates a way to calculate characters of irreducible highest weight modules over a reductive Lie algebra $\mathfrak{g}$ in terms of characters of Verma modules. The conjecture was proved by Beilinson--Bernstein and Kashiwara--Brylinski by their localization theorems. In the localization theorem, simple modules correspond to some IC sheaves, Verma modules correspond to some standard sheaves, and the parameter $q$ corresponds to the Tate twist $(1)$, $v=q^{\frac{1}{2}}$ corresponds to $(\frac{1}{2})$.

My naive understanding is that the localization theorem is the key in this problem; the deformation parameter $q$ is the shadow of some ``lifted mixed geometry''.

I wonder if there are more examples where quantum $q$ equals to arithmetic $q$, either in a similar (localization theorem, mixed geometry) or different flavor.