The geometry of stable lattices in Bruhat-Tits buildings
Appears in collection : AGCT - Arithmetic, Geometry, Cryptography and Coding Theory / AGCT - Arithmétique, géométrie, cryptographie et théorie des codes 2023
Let $K$ be a discretely valued field with ring of integers $R$ and let $d$ be a positive integer. Then the rank $d$ free $R$-submodules of $K^{d}$ (called $R$-lattices) are the $0$-simplices of an infinite simplicial complex called a Bruhat-Tits building. If $O$ is an order in the ring of $d\times d$ matrices over $K$, then the collection of lattices that are also $O$-modules (called $O$-lattices) is a non-empty, bounded and convex subset of the building. Determining what these subsets are is in general a difficult question. I will report on joint work with Yassine El Maazouz, Gabriele Nebe, Marvin Hahn, and Bernd Sturmfels describing the geometric features of the set of $O$-lattices for some particular orders. If time permits, I will also define spherical codes in Bruhat-Tits buildings and show how these fit in this framework and how they give rise to codes of submodules over chain rings.
