On Weight 2 Level N Newforms Congruent to Eisenstein Series When N Is Four Primes

Fix a prime $p\geq 5$. In this thesis we study the modularity problem of the Galois representation $\bar{\rho} = \mathbbm{1} \oplus \bar{\chi}$, where $ \bar{\chi}$ is a cyclotomic character mod $p$ and $\mathbbm{1}$ is the trivial one. Assuming $N$ is a product of four distinct primes and $p\nmid N$, we give a sufficient condition on $N$ such that there exists weight $2$ level $N$ newforms whose associated mod $p$ representation is isomorphic to $\bar{\rho}$. \\Our method is based on level-rising techniques using the geometry of Jacobian varieties of Shimura curves (see e.g. \cite{yoo2019non}) and pseudodeformation theory (see e.g. \cite{wake2021eisenstein}).