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I am looking for an example of a prime, p, for which there exists two $p$-ordinary rational elliptic curves $E$, $F$ for which, at every prime $l$ not dividing $N=p \operatorname{Cond}(E) \operatorname{Cond}(F)$:

$$\#E(\mathbb{F}_l) \equiv \#F( \mathbb{F}_l) \mod p $$

Such a congruence could be detected by a Hida Family having two weight $2$ modular forms $f,g,$ whose fields of fourier coefficients $K_f=K_g=\mathbb{Q}$. What I would prefer is exact equations of $E$ and $F$ (say, on www.lmfdb.org).

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  • $\begingroup$ I presume you wanted $\#E(\mathbb{F}_{l}) \equiv \#F(\mathbb{F}_{\ell})$, so I made an edit. $\endgroup$ – Jeremy Rouse Dec 6 '17 at 1:09
  • $\begingroup$ We have $\# E(\mathbb{F}_{\ell}) = \# F(\mathbb{F}_{\ell})$ for almost every $\ell$ if and only if $E$ and $F$ are isogenous. Most of the examples I describe below are cases where $E$ and $F$ are not isogenous, though (including the example for $p = 17$). $\endgroup$ – Jeremy Rouse Dec 6 '17 at 14:12
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There are many such curves. For $p = 2$, one can just take two curves $E$ and $F$ with $E : y^{2} = f(x)$ and $F : y^{2} = g(x)$ where $f(x)$ and $g(x)$ define the same number field. Also, if $E$ and $F$ are isogenous, then the above statement is true for any $p$ (but the corresponding modular forms are the same then).

Rubin and Silverberg have a couple of papers (see the MathSciNet reference here) that handle $p = 3$ and $5$. In particular, for these $p$, given an elliptic curve $E/\mathbb{Q}$, they construct a one-parameter family of elliptic curves $E_{t}$ so that $E[p]$ and $E_{t}[p]$ are isomorphic as Galois modules. (This works because such $E_{t}$ are parametrized by a twist of $X(p)$, and $X(p)$ has genus zero for $p = 3$ and $5$).

For larger $p$, there are still pairs (but not quite as many), and the best place to look is in this paper of Tom Fisher. In particular, the largest known example is with $p = 17$, which I mentioned in my answer to the question here. This example is a pair of $17$-ordinary elliptic curves that have $\# E(\mathbb{F}_{\ell}) \equiv \# F(\mathbb{F}_{\ell}) \pmod{17}$ for all primes $\ell$.

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