# $p$-adic spectral problem?

Warning: this is a very very naive question.

Spectral problem is one of the important subjects in the theory of automorphic forms (as I believe). If $$\Gamma$$ is a discrete subgroup of $$\mathrm{SL}_{2}(\mathbb{R})$$, then what is a spectrum of $$\Gamma\backslash\mathcal{H}$$, i.e. eigenfunctions on the quotient $$\Gamma\backslash \mathcal{H}$$ with respect to Laplace operators on it. As I know, this is a hard problem, such as Selberg's 1/4 conjecture isn't solved yet but solved for some cases.

We can consider the Laplacian operator as an element of $$U(\mathfrak{gl}_{2}(\mathbb{R}))$$, and we can apply representation theory of $$\mathfrak{gl}_{2}(\mathbb{R})$$ to study automorphic forms. Also, any (compactified) quotient $$\Gamma\backslash\mathcal{H}$$ is an algebraic curve over $$\mathbb{C}$$, and if $$\Gamma$$ is a congruence subgroup (for example, $$\Gamma = \Gamma_{0}(N)$$), then we even know that the curve is defined over $$\mathbb{Q}$$ or a number field. Although I don't know much about $$D$$-modules, I believe that we can think the differential operators (weight raising/lowering operators and Laplacian operator) can be interpreted purely algebraically, so that we can study these things algebraically.

My question is: can we do the similar thing over $$p$$-adic fields? Is it possible to use the modular curves over $$\mathbb{Q}_{p}$$ (maybe quotient of $$p$$-adic upper half plane by some $$p$$-adic discrete groups...) and $$D$$-modules on it to formulate spectral problem over $$p$$-adic fields? If my idea is correct, are there any known result for this direction?