To be or not to be local

Abstract

Let $p$ be a prime number and $K$ a finite unramified extension of ${\mathbf{Q}}_p$. For a smooth representation $\ddot{I}€$ of ${\mathrm{GL}}_2(K)$ occurring in some Hecke eigenspace of the mod $p$ cohomology of a Shimura curve, we explore different strategies (inspired by the case $K={\mathbf{Q}}_p$) to attack the locality question: does $\ddot{I}€$ depend only on the underlying $2$-dimensional representation $\overline{\ddot{I}}$ of $\mathrm{Gal}(\overline{K}/K)$? In particular when $[K:{\mathbf{Q}}_p]=2$, crucially using perfectoid geometry, we associate to $\overline{\ddot{I}}$ an infinite-dimensional mod $p$ smooth representation of $\left(\begin{array}{cc}{K}^{\times }& K\\ 0& 1\end{array}\right)$ which we hope is the restriction to $\left(\begin{array}{cc}{K}^{\times }& K\\ 0& 1\end{array}\right)$ of the (irreducible) supersingular subquotient of $\ddot{I}€$.