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?



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