  The modularity of elliptic curves always intrigues number theorists.
Recently, Thorne had proved a marvelous result that for a prime $ p $, every
elliptic curve defined over a $ p $-cyclotomic extension of $ \mathbb{Q} $ is
modular. The method is to use some automorphy lifting theorems and study
non-cusp points on some specific elliptic curves by Iwasawa theory for elliptic
curves. Since the modularity of elliptic curves over real quadratic was proved,
one may ask whether it is possible to replace $ \mathbb{Q} $ with a real
quadratic field $ K $. Following Thorne's idea, we give some assumptions first
and prove the modularity of elliptic curves over the $\mathbb{Z}_p$-extension
of some real quadratic fields.
