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

  1. 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 $f$?

  2. 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 $E'$?

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    $\begingroup$ Problem 2 is surely too hard, because it asks for a second rational point on the twist of ${\rm X}(p)$ by $E[p]$; once $p \geq 7$ that's a curve of genus $ > 1$ that already has one rational point, so there's no local obstruction to the existence of another rational point, but typically no such point exists. (For $p=2,3,5$ there are infinitely many such $E'$: for these small $p$, the curve ${\rm X}(p)$ has genus zero, so any twist with a rational point is a rational curve has infinitely many rational points.) $\endgroup$
    – Noam D. Elkies
    Commented Jan 10, 2022 at 2:13
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    $\begingroup$ Tom Fisher has a paper "Explicit moduli spaces for congruences of elliptic curves" for $p<12$ and one for $p=13$ and one for $17$, the last containing a strong conjecture as when elliptic curves could be congruent modulo $p\geq 17$ strengthening the Frey-Mazur conjecture. $\endgroup$
    – Chris Wuthrich
    Commented Jan 10, 2022 at 9:38
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    $\begingroup$ See also mathoverflow.net/questions/158702/… $\endgroup$
    – Alex B.
    Commented Jan 10, 2022 at 10:29
  • $\begingroup$ In 2, do you mean to ask for $E'$ to be not isomorphic/isogenous to $E$? Because I don't see why you couldn't take $E'=E$. $\endgroup$
    – Wojowu
    Commented Jan 10, 2022 at 15:24

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  1. 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).

  2. 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).

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