English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For a local field $F$ and an Artinian local coefficient ring $\Lambda$ withthe same positive residue characteristic $p$ we define, for any $e\in{\mathbbN}$, a category ${\mathfrak C}^{(e)}(\Lambda)$ of ${\rm GL}_2(F)$-equivariantcoefficient systems on the Bruhat-Tits tree $X$ of ${\rm PGL}_2(F)$. There isan obvious functor from the category of ${\rm GL}_2(F)$-representations over$\Lambda$ to ${\mathfrak C}^{(e)}(\Lambda)$. If $F={\mathbb Q}_p$ then${\mathfrak C}^{(1)}(\Lambda)$ is equivalent to the category of smooth ${\rmGL}_2({\mathbb Q}_p)$-representations over $\Lambda$ generated by theirinvariants under a pro-$p$-Iwahori subgroup. For general $F$ and $e$ we showthat the subcategory of all objects in ${\mathfrak C}^{(e)}(\Lambda)$ withtrivial central character is equivalent to a category of representations of acertain subgroup of ${\rm Aut}(X)$ consisting of "locally algebraicautomorphisms of level $e$". For $e=1$ there is a functor from this category tothat of modules over the (usual) pro-$p$-Iwahori Hecke algebra; it is abijection between irreducible objects. Finally, we present a parallel ofColmez' functor $V\mapsto {\bf D}(V)$: to objects in ${\mathfrakC}^{(e)}(\Lambda)$ (for any $F$) we assign certain \'{e}tale$(\varphi,\Gamma)$-modules over an Iwasawa algebra ${\mathfrako}[[\widehat{N}^{(1)}_{0,1}]]$ which contains the (usually considered) Iwasawaalgebra ${\mathfrak o}[[{N}_{0}]]$. This assignment preserves finitegeneration.

Locally algebraic automorphisms of the ${\rm PGL}_2(F)$-tree and ${\mathfrak o}$-torsion representations