Explicit Serre weights for GL₂ via Kummer theory

2022/07/01 by Bartlett, Robin, Steinmetz, Misja F. A.
#11F80 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2207.00402

Abstract

We give an explicit formulation of the weight part of Serre's conjecture for GL₂ 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.

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• Crystalline extensions and the weight part of Serre's conjecture
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• The weight part of Serre's conjecture over CM fields