3-divisibility of Manin constant for elliptic curves with 3-torsion

Question

Let $E/\mathbb{Q}$ be an elliptic curve with $E(\mathbb{Q}) \cong \mathbb{Z}/3\mathbb{Z}$ (not necessarily $\Gamma_0$-optimal). Does $3$ necessarily divide one of: the Manin constant (not necessarily $1$), the Tamagawa product, and the size of the Tate-Shafarevich group? I checked this for all curves in the LMFDB with conductor less than 10,000 and can't find any counter-examples.

Here is some context. The Birch–Swinnerton-Dyer quotients for rank zero elliptic curves $E/\mathbb{Q}$ generally have square-free denominators, due to Proposition 1.1 in Lorenzini's paper that classifies possible cancellations between the torsion and Tamagawa numbers. There are exceptions, notably when $E(\mathbb{Q}) \cong \mathbb{Z}/3\mathbb{Z}$, where Lemma 2.26 gives an infinite family where the Tamagawa product is $1$, or when $E(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z}$, which I'm not interested in. However, adding the size of the Tate-Shafarevich group into the picture makes the cancellation hold for many almost semistable elliptic curves with $E(\mathbb{Q}) \cong \mathbb{Z}/3\mathbb{Z}$, as proved in Theorem 3.1 in Melistas's paper. Moreover, adding the Manin constant into the picture seems to make the cancellation hold unconditionally, as noted in Example 3.8 (and Example 2.6, but I didn't check LMFDB for this).

I don't know how to prove results on the Manin constant, so this was just something I observed experimentally. Cesnavicius proved that the Manin constant of a semistable $\Gamma_0$-optimal elliptic curve is $1$, but I can't find any existing literature proving that the Manin constant is not $1$ for families of elliptic curves that are not $\Gamma_0$-optimal. I'm trying to show that the special L-value (times the Manin constant) has square-free denominators, so a direct way to prove this without assuming BSD would be ideal.

Answer 1

Perhaps there is an elementary answer after all. Here is my attempt at a partial answer assuming BSD.

Let $c_0(E)$ denote the Manin constant of $E$, and let $L(E)$ denote the special L-value of $E$ divided by its real period. For any odd prime $p$ not dividing the conductor of $E$, the Hecke operator $T_p$ acts on the space of modular symbols of $E$, and by pairing up the periods (similar to Proposition 7 of Wiersema–Wuthrich), it can be shown that $c_0(E)L(E)|E(\mathbb{F}_p)|$ is integral. There are stronger integrality statements replacing $|E(\mathbb{F}_p)|$ by the torsion subgroup in Proposition 4.6 of Agashe, but it's only stated for optimal quotients.

If $3$ does not divide $c_0(E)$, then $L(E)|E(\mathbb{F}_p)|$ is integral at $3$, so by BSD it suffices prove that $|E(\mathbb{F}_p)|$ is divisible by $3$ but not $9$, for some odd prime $p$. This argument does work when $E$ has good reduction at $3$, in which case the Hasse bound forces $|E(\mathbb{F}_3)|$ to be $3$ or $6$. I wonder if the elementary integrality argument can be adapted to work with the Hecke operator $U_3$ when $E$ has bad reduction at $3$.

In general, we would have to inspect the mod $9$ Galois image of $E$ to obtain a Frobenius at $p$ for which $|E(\mathbb{F}_p)|$ is divisible by $3$ but not $9$. I have looked at all 42 possible $3$-adic Galois images of $E$ (given the rational $3$-isogeny coming from torsion), thanks to the classification in Corollary 1.3.1 and 12.3.3 in Rouse–Sutherland–Zureick-Brown. In all cases except the subgroup labels 9.72.0.1, 9.72.0.5, and 9.72.0.12, there are primes $p$ that work, but in these three cases (e.g. the isogeny class with Cremona label 54b) I am stuck.

Any suggestions would be very much appreciated!

EDIT: I think I found an argument involving a case-bash that finishes off the proof, but I would still prefer a nicer conceptual proof to this, so I'll not accept my own answer to await a better argument.

The subgroup label 9.72.0.5 has $E(\mathbb{Q}) \cong \mathbb{Z}/9\mathbb{Z}$ (since its mod $9$ image fixes a line) and the subgroup label 9.72.0.12 has $E(\mathbb{Q}) \cong 1$ (since its mod $3$ image does not fix anything), so we can ignore these cases. The remaining subgroup label 9.72.0.1 has mod $3$ image that is Cartan, and I claim that in this case there is some prime $p$ such that the Tamagawa number $c_p$ is divisible by $3$.

Now the elliptic curves with $c_p = 1$ and non-trivial torsion are completely classified thanks to the work of Lorenzini (aforementioned) and Theorem 1.2 and Theorem 5.2 in Barrios–Roy. For elliptic curves with $3$-torsion, by following the argument in the proof of Theorem 1.2, one could read from the table in Theorem 3.5 that $c_p$ is divisible by $3$ if $E$ is not isomorphic to the elliptic curve $y^2 + a^3xy \pm a^6y = x^3$ for any $a \in \mathbb{Z}$ (the other possibility $y^2 + ay^2 = x^3$ has CM, and thanks to the Rouse–Sutherland–Zureick-Brown classification, CM mod $9$ images have primes $p$ such that $|E(\mathbb{F}_p)|$ is divisible by $3$ but not $9$). Thus it suffices to prove that if $E$ is isomorphic to an elliptic curve of this form, which by a change of variables is just $y^2 + xy \pm \tfrac{1}{a^3}y = x^3$, then it cannot have mod $3$ image that is Cartan.

Finally, mod $3$ images can be read off from the $j$-invariant thanks to Theorem 1.2 in Zywina. By a simple calculation with $t = \tfrac{27}{a^3 - 27}$, our elliptic curve has $j$-invariant of the form $J_3(t)$, so Theorem 1.2 establishes that it has mod $3$ image that is Borel, which is in particular not Cartan.

I suppose this final argument isn't that elementary after all.