# How can I determine in practice whether two elliptic curves over $\mathbb{Q}$ have isomorphic $p$-torsion?

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ with good, ordinary reduction at an odd prime $p$. I'm wondering how to determine whether $E_1[p]$ and $E_2[p]$ are isomorphic $\mathbb{F}_p[G_\mathbb{Q}]$-modules.

It is my understanding that I can check this by looking for a congruence mod $p$ between the associated modular forms, but I'm not exactly sure what constitutes a congruence. I guess that it should mean that almost all of the numbers $a_\ell$ for the two curves (for $\ell\neq p$ a prime of good reduction for both $E_1$ and $E_2$) are congruent mod $p$. But this confuses me for the following reason. These $a_\ell$ are the traces of Frobenii on the $p$-adic Tate module, and I assume the reason this idea should work is that if they are (almost) all congruent mod $p$, then the $\mathbb{F}_p$-representations should be isomorphic by considerations with Chebotarev density (assuming these representations are semisimple, a fact which I think\hope is true but for which I have no reference).

If this is indeed what is meant by a congruence, then how could I check it in practice? I can look up my curves in Cremona's tables and look at as many of coefficients of the $q$-expansions of the modular forms as I want, but how many do I have to look at before I conclude that the congruence holds?

Disclaimer: I am very much new to computational stuff, so if I have said something naive or borderline ridiculous, I apologize. I'm more accustomed to working sort of...theoretically.

## 1 Answer

I believe what you are looking for is called "the Sturm bound". See, for example, William Stein's "Modular forms, a computational approach" (which is freely available online). In particular, read Section 9.4, on congruences between newforms. Sturm's theorem is Theorem 9.18 in that section, and the applications to congruences appear in Corollaries 9.19 and 9.20, and Theorems 9.21 and 9.22.

If you are new to computing with modular forms, Stein's book is a very nice reference on the subject.

• Thanks very much for the specific references. I will definitely check out those sections. – Keenan Kidwell Apr 4 '12 at 0:45
• Okay, so I looked those results up, and I have a sort of follow-up question. The results essentially tell me that if the prime index Fourier coefficients of two forms are congruent mod some maximal ideal for prime indices less then or equal to an explicit bound, then all the coefficients are congruent. But it doesn't address what happens when there may be finitely many primes where the coefficients are not congruent. For instance, in a paper I'm reading the author asserts that the elliptic curves $E_1$ (Cremona's 52a1) and $E_2$ (Cremona's 364a1) have isomorphic mod $5$ representations, and – Keenan Kidwell Apr 4 '12 at 3:12
• claims that this can be seen by looking at the $q$-expansions of the associated modular forms and observing that neither curve admits a rational $5$-isogeny. I've looked at the $q$-expansions, and many of the prime index coefficients are congruent mod $5$, but not all of them. It's possible the fact about $5$-isogenies somehow allows one to ignore the exception coefficients, but I don't understand why. – Keenan Kidwell Apr 4 '12 at 3:14
• I guess maybe the non-existence of a $5$-isogeny implies that the $5$-torsion representations are irreducible... – Keenan Kidwell Apr 4 '12 at 18:41
• @KeenanKidwell: For what primes $p$ are the $a_p$ coefficients not congruent mod $5$? What primes divide the conductors of $E_1$ and $E_2$? What is the set $I$ in Theorem 9.22 of Stein's book? – Álvaro Lozano-Robledo Apr 5 '12 at 17:05