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

In this article, we prove the remaining open cases of the Fontaine-Mazur conjecture on two-dimensional regular Galois representations over $\text{Gal}(\bar{\text{Q}}\mathrm{/}\text{Q})$ 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.