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 $E'$?
Best Answer
Given an eigencuspform $f$ of weight $k≥2$ (so in particular 2) and $p$ a prime of ordinary reduction (in particular good ordinary reduction), there is always a Hida family passing through $f$. This is for instance Theorem I of Galois representations into $\operatorname{GL}_{2}(\mathbb Z_{p}[[X]])$ attached to ordinary cusp forms by H.Hida (Inventiones Mathematicae, 1986).
As Noam Elkies writes in comment, the second question appears at present to be hopelessly hard. It is generally believed that all examples of elliptic curves congruent modulo a large prime $p$ should be very restricted but I don't think we are anywhere near a proof.
In summary, the answer to your first question is the best possible (positive, with well-documented references) while the answer to your second question is the worse possible (probably negative but nobody knows).