5 problems

Let $A$ be the optimal elliptic curve associated to the weight-$2$ newform $f$ of level $N$, and for each prime $p\mid N$ let $c_p(A)$ be its Tamagawa number. Agashe's Tamagawa-lev…

Let $\rho:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}_{2}(k)$ be an irreducible representation valued in a finite field $k$ of characteristic…

Let $p$ be a prime, let $N=N^+Q$ with $N$ square-free, and let $f$ be a weight two newform of level $\Gamma_0(N)$ with associated residual representation $\overline{\rho}_f$. Suppo…

Let $p$ be a prime with $p\geq 5$, and let $f$ be a newform of weight two and square-free level $N=a\cdot q\cdot b$, where $q$ is prime and $(N,p)=1$. Let $\lambda$ be the prime of…

Ribet-style level-lowering conjecture. Under these hypotheses, such an automorphic form $G$ exists at level with the prime factor $\operatorname{id}{p}$ removed.