On the Fontaine-Mazur conjecture for $p=3$

TL;DR

This paper proves the remaining cases of the Fontaine-Mazur conjecture for two-dimensional Galois representations at p=3, completing the conjecture for all odd primes in the regular case.

Contribution

It extends previous work by confirming the conjecture for all odd primes at p=3 using recent advances in p-adic Langlands correspondence and Galois deformation theory.

Findings

Confirmed the Fontaine-Mazur conjecture for all odd primes at p=3 in the regular case.

Built upon recent progress in p-adic Langlands correspondence and Galois deformation theory.

Provided a comprehensive proof completing the conjecture in the regular case.

Abstract

In this article, we prove the remaining open cases of the Fontaine-Mazur conjecture on two-dimensional regular Galois representations over $\Gal(\overline{\Q}/\Q)$ when $p=3$, hence concluding the conjecture in the regular case for all odd primes. Our result is a sequel to Pan's work, based on some recent progress on $p$-adic Langlands correspondence, Galois deformation theory and a potential pro-modularity result.