Integral structures in automorphic line bundles on the $p$-adic upper half plane

TL;DR

This paper constructs integral lattices in automorphic line bundles on the p-adic upper half plane, generalizing previous work to all integer weights, and explores their cohomological properties and applications to de Rham cohomology of certain algebraic curves.

Contribution

It extends Teitelbaum's construction of integral structures to arbitrary integer weights and analyzes their cohomology and monodromy in the p-adic setting.

Findings

Constructed GL_2(K)-equivariant integral lattices for all integer weights.

Proved degeneration of a Hodge spectral sequence for de Rham cohomology.

Showed the monodromy operator respects integral structures and is universal.

Abstract

Given an automorphic line bundle ${\mathcal O}_X(k)$ of weight $k$ on the Drinfel'd upper half plane $X$ over a local field $K$, we construct a ${\rm GL}_2(K)$-equivariant integral lattice ${\mathcal O}_{\widehat{\mathfrak X}}(k)$ in ${\mathcal O}_X(k)\otimes_K\widehat{K}$, as a coherent sheaf on the formal model $\widehat{\mathfrak{X}}$ underlying $X\otimes_K\widehat{K}$. Here $\widehat{K}/K$ is ramified of degree $2$. This generalizes a construction of Teitelbaum from the case of even weight $k$ to arbitrary integer weight $k$. We compute $H^*(\widetilde{\mathfrak{X}},{\mathcal O}_{\widehat{\mathfrak X}}(k))$ and obtain applications to the de Rham cohomology $H_{dR}^1(\Gamma\backslash X,{\rm Sym}_K^k({\rm St}))$ with coefficients in the $k$-th symmetric power of the standard representation of ${\rm SL}_2(K)$ (where $k\ge0$) of projective curves $\Gamma\backslash X$ uniformized by $X$: namely, we prove the degeneration of a certain reduced Hodge spectral sequence computing $H_{dR}^1(\Gamma\backslash X,{\rm Sym}_K^k({\rm St}))$, we re-prove the Hodge decomposition of $H_{dR}^1(\Gamma\backslash X,{\rm Sym}_K^k({\rm St}))$ and show that the monodromy operator on $H_{dR}^1(\Gamma\backslash X,{\rm Sym}_K^k({\rm St}))$ respects integral de Rham structures and is induced by a "universal"{} monodromy operator defined on $\widehat{\mathfrak{X}}$, i.e. before passing to the $\Gamma$-quotient.