# Equivalent notions of congruence for elliptic curves over $\mathbb{Q}$

Let $$E_1$$ and $$E_2$$ be elliptic curves over $$\mathbb{Q}$$ and $$f_i$$ the eigencuspform of weight $$2$$ attached to $$E_i$$. Express $$f_1=\sum a_i q^i$$ and $$f_2=\sum b_i q^i$$.

Suppose that the residual Galois representations $$E_1[p]\simeq E_2[p]$$. Note that $$E_1$$ and $$E_2$$ have bad reduction at the same primes.

It is easy to see that $$a_l\equiv b_l\mod{p}$$ at a prime $$l$$ of good reduction for both elliptic curves (since the residual representations are unramified at $$l$$). Does it follow that $$a_l\equiv b_l\mod{p}$$ at a prime $$l$$ of bad reduction for both elliptic curves. If so how do we show this? Perhaps this comes down to showing that $$E_1$$ has the same kind of bad reduction at $$p$$ as $$E_2$$. I know that if $$E_1$$ has additive reduction then so does $$E_2$$, but I'm not sure about distinguishing between split multiplicative and non-split multiplicative reduction.

• Do you mean to impose some conditions on $p$ or the residual representation? In general they don't have the same primes of bad reduction. If $p = 2$ then quadratic twisting gives you many counterexamples. Curves with $E[3] \simeq \mathbb{Z}/3 \times \mu_3$ give you many counterexamples as well. – Ari Shnidman Sep 3 '19 at 20:40
• @AriShnidman Let me assume that $p>2$ and that the residual representation $E_1[p]\simeq E_2[p]$ is irreducible. – user130124 Sep 3 '19 at 21:03

I found myself wondering the same thing a couple of weeks ago. Even with the restriction that $$p > 2$$ and the residual representation is irreducible, it does not follow that $$E_{1}$$ and $$E_{2}$$ have the same primes of bad reduction, nor that $$a_{\ell}(E_{1}) \equiv a_{\ell}(E_{2}) \pmod{p}$$ for primes of bad reduction.
For example, take $$p = 5$$ and $$E_{1} : y^{2} + xy + y = x^{3} + 4x - 6$$ (aka $$X_{0}(14)$$). The mod $$5$$ Galois representation attached to $$E_{1}$$ is surjective. For any elliptic curve $$E$$, Rubin and Silverberg write down an isomorphism $$X_{E}(5) \cong \mathbb{P}^{1}$$, the modular curve parametrizing elliptic curves $$F$$ so that $$F[5]$$ and $$E[5]$$ are isomorphic via and isomorphism respecting the Weil pairing. For $$E = E_{1}$$, one can take $$E_{2} : y^{2} + xy = x^{3} - x^{2} - 4492x + 126416$$. This curve has $$E_{1}[5] \cong E_{2}[5]$$, but $$E_{2}$$ has bad reduction at $$5$$ and $$E_{1}$$ has good reduction at $$5$$. It is not surprising that one can fail to have $$a_{\ell}(E_{1}) \equiv a_{\ell}(E_{2}) \pmod{p}$$ when one of $$E_{1}$$ and $$E_{2}$$ has good reduction at $$\ell$$ and the other doesn't.
One can also have $$a_{\ell}(E_{1}) \not\equiv a_{\ell}(E_{2}) \pmod{p}$$ when both curves have bad reduction at $$\ell$$. Looking in the same family (of curves directly $$5$$-congruent to $$E_{1}$$) one can find curves $$E_{2}$$ and $$E_{3}$$ that have multiplicative reduction at $$199$$ where one curve has $$a_{199}(E_{2}) = -1$$ and the other has $$a_{199}(E_{3}) = 1$$. The examples I found are a bit horrendous: $$E_{2} : y^2 + xy = x^3 - x^2 - 183192520591565859491828595856323163822228663229886627562x + 1062123179152064591727007023640066384957014639073044998393946451273297951607053468340$$ and $$E_{3} : y^2 + xy + y = x^3 - 187739104428369548177010660620161617529156118721103898180850571x + 1108087782673654453656602976287807709340970590450649179565293267500311360376488672021879300982.$$
• Thank you, yes I was actually interested in $l\neq p$ in which case both curves have good reduction at the same primes, and the example you provided shows that one cannot distinguish split and non-split multiplicative reduction at a prime $l\neq p$. – user130124 Sep 4 '19 at 1:15
Note that a conjecture attributed to Frey and Mazur says that for each elliptic curve $$E_1$$, there exists a constant $$C$$ depending on $$E_1$$, such that for every prime $$p>C$$ and for every elliptic curve $$E_2$$ defined over $$\mathbb{Q}$$ with $$E_1[p]\simeq E_2[p]$$ as Galois modules, one has $$E_1$$ isogenous to $$E_2$$. In particular, $$E_1$$ and $$E_2$$ have the same $$L$$-functions and hence the same coefficients $$a_\ell$$ (including at the bad primes).
It is even conjectured that $$C$$ can be taken independently of $$E_1$$. Moreover there is no known pair of non-isogenous elliptic curves $$(E_1,E_2)$$ such that $$E_1[p]\simeq E_2[p]$$ for some prime $$p>17$$. If you want to learn more about this conjecture you can have a look at the recent preprint by Cremona and Freitas "Global methods for the symplectic type of congruences between elliptic curves": https://arxiv.org/abs/1910.12290