Integral structures in automorphic line bundles on the \(p\)-adic upper half plane (Q1889452)

Let \(X\) be the Drinfel'd upper half plane over a local field \(K\) (with the ring of integers \(O_K\) and the residue field \(\mathbb{F}\), and \(\widehat K/K\) a ramified extension of degree 2. The author constructs a coherent sheaf \(O_{\widehat{\mathcal X}}(k)\) on the formal model \(\widehat{\mathcal X}\) underlying \(X\bigotimes_KR\) (generalizing a construction of \textit{J. T. Teitelbaum} [Invent. Math. 113, No. 3, 561--580 (1993; Zbl 0806.11027)]), and computes \(H^*(\widetilde{\mathcal X},O_{\widehat{\mathcal X}}(k))\), where \(\widetilde{\mathcal X}= \widehat{\mathcal X}\otimes_{O_R}\mathbb{F}\). Applications include: (i) the degeneration of a certain reduced Hodge spectral sequence computing\break \(H^1_{dR}(\Gamma\setminus X,\text{Sym}^k_K(St))\), where \(\Gamma\setminus X\) is a projective curve uniformized by \(X\) and \(\text{Sym}^k_K(St)\) denotes the \(k\)th symmetric power of a standard representation of \(\text{SL}_2(K)\) [conjectured by \textit{P. Schneider}, Math. Ann. 293, No. 4, 623--650 (1992; Zbl 0774.14022)]; (ii) a new proof of the Hodge decomposition of \(H^1_{dR}(\Gamma\setminus X,\text{Sym}^k_K(St))\); (iii) a proof that the monodromy operator on \(H^1_{dR}(\Gamma\setminus X,\text{Sym}^k_K(St))\) respects integral de Rham structures.