On congruences mod ${\mathfrak{p}}^m$ between eigenforms and their attached Galois representations

I. Chen, I. Kiming, J. B. Rasmussen

Given a prime $p$ and cusp forms $f_1$ and $f_2$ on some ${\Gamma }_1(N)$ that are eigenforms outside $Np$ and have coefficients in the ring of integers of some number field $K$, we consid…