Families of two dimensional modular $(\varphi,\Gamma)$-modules

TL;DR

This paper constructs explicit integral structures for rank two e9tale Lubin-Tate modules over finite fields, revealing new combinatorial structures and linking Serre weights to the geometry of the moduli stack.

Contribution

It provides explicit algebraic families of integral structures for two-dimensional e9tale Lubin-Tate modules and uncovers their combinatorial and geometric properties.

Findings

Constructed integral structures for all rank two e9tale Lubin-Tate modules.

Revealed new combinatorial structures of the moduli stack.

Connected Serre weights to the geometry of the moduli stack.

Abstract

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 \'{e}tale Lubin-Tate $(\varphi,{\mathcal O}_F^{\times})$-modules over $k$. We construct algebraic families of such integral structures and show that these comprehensively reflect the degeneration behaviour of $(\varphi,{\mathcal O}_F^{\times})$-modules. These results reveal new combinatorial structures of the moduli stack of $(\varphi,{\mathcal O}_F^{\times})$-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.