Explicit Serre weights for two-dimensional Galois representations

TL;DR

This paper proves an explicit version of the Buzzard--Diamond--Jarvis conjecture for two-dimensional Galois representations, confirming its equivalence to the original conjecture and extending previous results for odd primes.

Contribution

It establishes the explicit form of the Buzzard--Diamond--Jarvis conjecture and proves its equivalence to the original conjecture for all primes.

Findings

Proved the explicit Buzzard--Diamond--Jarvis conjecture for two-dimensional Galois representations.

Confirmed the equivalence between the explicit and original conjectures.

Extended previous results to include all primes, not just odd primes.

Abstract

We prove the explicit version of the Buzzard--Diamond--Jarvis conjecture formulated by Diamond--Demb\'el\'e--Roberts. More precisely, we prove that it is equivalent to the original Buzzard--Diamond--Jarvis conjecture, which was proved for odd primes (under a mild Taylor--Wiles hypothesis) in earlier work of the third author and coauthors.