On irreducible supersingular representations of $\mathrm{GL}_{2}(F)$

TL;DR

This paper investigates the structure of supersingular representations of 2(F) over non-archimedean local fields, demonstrating the existence of infinitely many non-isomorphic irreducible admissible quotients using cyclic diagrams.

Contribution

It introduces cyclic diagrams as a tool to analyze supersingular modules and proves the existence of infinitely many irreducible quotients for these modules.

Findings

Existence of infinitely many non-isomorphic irreducible admissible quotients

Introduction of cyclic diagrams as an analytical tool

Application to 2(F) representations over fields with residual characteristic p>3

Abstract

Let $F$ be a non-archimedean local field of residual characteristic $p>3$ and residue degree $f>1$. We study a certain type of diagram, called \emph{cyclic diagrams}, and use them to show that the universal supersingular modules of $\mathrm{GL}_{2}(F)$ admit infinitely many non-isomorphic irreducible admissible quotients.