The modularity of elliptic curves over all but finitely many totally real fields of degree 5

We study the finiteness of low degree points on certain modular curves and their Atkin--Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree $5$. On the way, we prove a criterion for the finiteness of rational points of degree $5$ on a curve of large genus over a number field using the results of Abramovich--Harris and Faltings on subvarieties of Jacobians.Comment: 23 pages. Final version, accepted for publication in 'Research in Number Theory'. To the newest version, we added the ancillary files which were attached to v4 but not to v5. The attachment contains data on modular forms (of level 105, 315, and 735) which are generated by LMFD