Explicit Serre weights for GL_2 via Kummer theory

TL;DR

This paper provides an explicit, Kummer theory-based formulation of the weight part of Serre's conjecture for GL_2, simplifying previous approaches by avoiding p-adic Hodge theory and reducing combinatorial complexity.

Contribution

It introduces a new explicit formulation of Serre's conjecture weights for GL_2 using Kummer theory, connecting it to existing local class field theory results.

Findings

Explicit description of crystalline extension reductions via Kummer theory

Recovery of previous formulations using explicit reciprocity laws

Simplification of combinatorial aspects in weight calculations

Abstract

We give an explicit formulation of the weight part of Serre's conjecture for GL_2 using Kummer theory. This avoids any reference to p-adic Hodge theory. The key inputs are a description of the reduction modulo p of crystalline extensions in terms of certain "G_K-Artin-Scheier cocycles" and a result of Abrashkin which describes these cocycles in terms of Kummer theory. An alternative explicit formulation in terms of local class field theory was previously given by Dembele-Diamond-Roberts in the unramified case and by the second author in general. We show that the description of Dembele-Diamond-Roberts can be recovered directly from ours using the explicit reciprocity laws of Brueckner-Shaferevich-Vostokov. These calculations illustrate how our use of Kummer theory eliminates certain combinatorial complications appearing in these two papers.