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Analytic Langlands correspondence was proposed by Etingof, Frenkel andKazhdan. On one side of this correspondence there are certain operators on$L^2(\operatorname{Bun}_G)$, called Hecke operators, where$\operatorname{Bun}_G$ is the variety of stable $G$-bundles on $X$ and$L^2(\operatorname{Bun}_G)$ is a Hilbert space of square-integrablehalf-densities. The compactness conjecture says that Hecke operators arebounded and, moreover, compact. In arXiv:2106.05243 Etingof, Frenkel andKazhdan prove this and other conjectures in the case of$G=\operatorname{PGL}_2$ and $X=\mathbb{P}^1$ with parabolic structures. Weinvestigate the case of $G=\operatorname{PGL}_2$ and genus one curve overcomplex numbers with parabolic structures. We obtain an explicit formula forHecke operators and prove the compactness conjecture in this case.

Analytic Langlands correspondence for $\operatorname{PGL}_2(\mathbb{C})$ on a genus one curve with parabolic structures