Modularity theorems for abelian surfaces

We prove the modularity of a positive proportion of abelian surfaces over $\mathbf{Q}$. More precisely, we prove the modularity of abelian surfaces which are ordinary at $3$ and are $3$-distinguished, subject to some assumptions on the $3$-torsion representation (a “big image” hypothesis, and a technical hypothesis on the action of a decomposition group at $2$). We employ a 2-3 switch and a new classicality theorem (in the style of Lue Pan) for ordinary $p$-adic Siegel modular forms.