I'd like to ask two questions about congruences: one about modular forms and one about elliptic curves.

Suppose we are given a cusp form $f$ of weight $2$ and level $\Gamma_0(N)$. Given a good ordinary prime $p$ for $f$, does there always exist a Hida family passing through $f$? (I've heard that such a Hida family always exists when $p$ is a

*bad*prime, but does one exist if $p$ is*good ordinary*?) If such a Hida family doesn't always exist, are there additional conditions we can place on $f$ and $p$ that would ensure the existence of a Hida family passing through $f$?If $E/\mathbf{Q}$ is an elliptic curve and $p$ is a prime of good ordinary reduction, under what conditions can we find a curve $E'$ that is congruent to $E$ mod $p$? (i.e: such that their mod $p$ Galois representations are isomorphic?) Are there sufficient conditions we can place on $E$ and $p$ that ensure the existence of a curve $E'$?