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- W2287587339 abstract "For a local field $F$ and an Artinian local coefficient ring $Lambda$ with the same positive residue characteristic $p$ we define, for any $ein{mathbb N}$, a category ${mathfrak C}^{(e)}(Lambda)$ of ${rm GL}_2(F)$-equivariant coefficient systems on the Bruhat-Tits tree $X$ of ${rm PGL}_2(F)$. There is an 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 ${rm GL}_2({mathbb Q}_p)$-representations over $Lambda$ generated by their invariants under a pro-$p$-Iwahori subgroup. For general $F$ and $e$ we show that the subcategory of all objects in ${mathfrak C}^{(e)}(Lambda)$ with trivial central character is equivalent to a category of representations of a certain subgroup of ${rm Aut}(X)$ consisting of locally algebraic automorphisms of level $e$. For $e=1$ there is a functor from this category to that of modules over the (usual) pro-$p$-Iwahori Hecke algebra; it is a bijection between irreducible objects. Finally, we present a parallel of Colmez' functor $Vmapsto {bf D}(V)$: to objects in ${mathfrak C}^{(e)}(Lambda)$ (for any $F$) we assign certain '{e}tale $(varphi,Gamma)$-modules over an Iwasawa algebra ${mathfrak o}[[widehat{N}^{(1)}_{0,1}]]$ which contains the (usually considered) Iwasawa algebra ${mathfrak o}[[{N}_{0}]]$. This assignment preserves finite generation." @default.
- W2287587339 created "2016-06-24" @default.
- W2287587339 creator A5040373321 @default.
- W2287587339 date "2015-07-21" @default.
- W2287587339 modified "2023-09-27" @default.
- W2287587339 title "Locally algebraic automorphisms of the ${rm PGL}_2(F)$-tree and ${mathfrak o}$-torsion representations" @default.
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