arXiv subjects

Elmar Große-Klönne

Publications and source records attributed to Elmar Große-Klönne.

4 recordsLinked to original sources

arXiv2024

Families of two dimensional modular $(φ,Γ)$-modules

Let $F/{\mathbb Q}_p$ be a finite unramified extension, let $k$ be a finite extension of the residue field of $F$. We provide explicit constructions of integral structures for all rank two étale Lubin-Tate $(φ,{\mathcal O}_F^{\times})$-modules over $k$. We construct algebraic families of such integral structures and show that these comprehensively reflect the degeneration behaviour of $(φ,{\mathcal O}_F^{\times})$-modules. These results reveal new combinatorial structures of the moduli stack of $(φ,{\mathcal O}_F^{\times})$-modules, and allow us, in particular, to rederive the fact that the Serre weights assigned to a two dimensional ${\rm Gal}(\overline{F}/F)$-representation over $k$ can be read off from the geometry of the stack.

math.NT↗

arXiv2024

On $ψ$-lattices in modular $(φ,Γ)$-modules

Let $F/{\mathbb Q}_p$ be a finite field extension, let $k$ be a finite field extension of the residue field of $F$. Generalizing the $ψ$-lattices which Colmez constructed in étale $(φ,Γ)$-modules over $k[[t]][t^{-1}]$, we define, study and exemplify $ψ$-lattices in étale $(φ,Γ)$-modules over $k[[t_1,\ldots,t_d]][\prod_it_i^{-1}]$ for arbitrary $d\in{\mathbb N}$.

math.NT↗

arXiv2020

Supersingular Hecke modules as Galois representations

Let $F$ be a local field of mixed characteristic, let $k$ be a finite extension of its residue field, let ${\mathcal H}$ be the pro-$p$-Iwahori Hecke $k$-algebra attached to ${\rm GL}_{d+1}(F)$ for some $d\ge1$. We construct an exact and fully faithful functor from the category of supersingular ${\mathcal H}$-modules to the category of ${\rm Gal}(\overline{F}/F)$-representations over $k$. More generally, for a certain $k$-algebra ${\mathcal H}^{\sharp}$ surjecting onto ${\mathcal H}$ we define the notion of $\sharp$-supersingular modules and construct an exact and fully faithful functor from the category of $\sharp$-supersingular ${\mathcal H}^{\sharp}$-modules to the category of ${\rm Gal}(\overline{F}/F)$-representations over $k$.

math.NT↗

arXiv2018

A note on multivariable $(φ,Γ)$-modules

Let $F/{\mathbb Q}_p$ be a finite field extension, let $k$ be a field of characteristic $p$. Fix a Lubin Tate group $Φ$ for $F$ and let $Γ\times\cdots\timesΓ$ with $Γ={\mathcal O}_F^{\times}$ act on $k[[t_1,\ldots,t_n]][\prod_it_i^{-1}]$ by letting $γ_i$ (in the $i$-th factor $Γ$) act on $t_i$ by insertion of $t_i$ into the power series attached to $γ_i$ by $Φ$. We show that $k[[t_1,\ldots,t_n]][\prod_it_i^{-1}]$ admits no non-trivial ideal stable under $Γ$, thereby generalizing a result of Zábrádi (who had treated the case where $Φ$ is the multiplicative group). We then discuss applications to $(φ,Γ)$-modules over $k[[t_1,\ldots,t_n]][\prod_it_i^{-1}]$.

math.NT↗