Generic local deformation rings when $l \neq p$

TL;DR

This paper characterizes local deformation rings for generic mod l Galois representations over p-adic fields and proves a related conjecture in the tame case, advancing understanding in number theory and representation theory.

Contribution

It provides a detailed description of local deformation rings for generic mod l Galois representations when l ≠ p, connecting them to conjugacy classes in the dual group.

Findings

Explicit description of local deformation rings for generic mod l representations.

Proof of the l ≠ p Breuil–Mézard conjecture in the tame case.

Connection established between deformation rings and conjugacy classes in the dual group.

Abstract

We determine the local deformation rings of sufficiently generic mod $l$ representations of the Galois group of a $p$-adic field, when $l \neq p$, relating them to the space of $q$-power-stable semisimple conjugacy classes in the dual group. As a consequence we give a local proof of the $l \neq p$ Breuil--M\'{e}zard conjecture of the author, in the tame case.