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 naïve 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.