Finite Length for Unramified $GL_2$: Beyond Multiplicity One
Lucrezia Bertoletti
TL;DR
This work proves that smooth mod $p$ representations of $\mathrm{GL}_2(K)$, arising from the cohomology of Shimura curves, have finite length under genericity assumptions even without multiplicity-one at tame level. The authors develop a local-to-global strategy via diagrammatic parametrizations $D(\overline{\rho})$, and $(\varphi,\Gamma)$-module techniques, establishing Cohen–Macaulay properties of associated graded modules and controlling $\mathfrak{m}_{K_1}^{2}$-torsion. They show that subquotients are generated by their $\mathrm{GL}_2(\mathcal{O}_K)$-socle, yielding explicit decompositions into principal-series and supersingular pieces, with bounds on length depending on a multiplicity parameter $r$ and residue field degree $f$. The results culminate in a global finiteness statement for $\pi(V^{v})$, valid for $\max\{12,2f+1\}$-generic $\overline{\rho}$, extending prior work that required a multiplicity-one hypothesis. Overall, the paper provides a robust framework to analyze mod $p$ GL$_2$-representations in Shimura-curve settings, combining diagrammatic methods, $\varphi$-$\Gamma$-modules, and Cohen–Macaulayness to obtain finite length and structural results with potential applications to $p$-adic and automorphic programs.
Abstract
Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. Building on recent work of Breuil, Herzig, Hu, Morra and Schraen, we study the smooth mod $p$ representations of $\mathrm{GL}_2(K)$ appearing in a tower of mod $p$ Hecke eigenspaces of the cohomology of Shimura curves, under mild genericity assumptions but notably no multiplicity one assumption at tame level, and prove that these representations are of finite length, thereby extending a previous result of the aforementioned authors.