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Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$. Let $p\ge11$ be a prime of good ordinary reduction for $E$ and assume that $p$ does not divide the degree of a minimal modular parametrization $\varphi_E:X_0(N)\to E$.

It's known by a work of Mazur that in this setting $p$ does not divide the Manin constant of $E$.

Is it also true that $p$ does not divide the product of the Tamagawa numbers $C=\prod_{\ell|N}c_\ell(E)$?

If the answer is affirmative, can anyone give me a reference? Conversely, does anyone know a counterexample? Are there conditions implying the non divisibility?

Sorry if the question is silly.

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  • $\begingroup$ A number of people commented that there is a counterexample for $p=5$, which is of course not quite what you're asking about. I'll just note that the Tamagawa number of an elliptic curve over a dvf is essentially the valuation of the minimal discriminant (and thus $\le 12$) if it has split multiplicative reduction and something else (not a large prime) otherwise. So we're really talking about $p=11$ on the nose here and having sage search through (some fraction of) the Cremona database didn't find anything with tamagawa number divisible by 11. $\endgroup$
    – stankewicz
    Dec 17, 2013 at 16:32
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    $\begingroup$ @stankewicz If $E$ has split multiplicative reduction at $\ell$, then you are correct that $c_\ell(E)=\text{ord}_\ell(\mathcal{D}_E)$, where $\mathcal{D}_E$ is the minimal discriminant. But this valuation can be arbitrarily large, it is not $\le12$ as you assert. So the question is not only about $p=11$. $\endgroup$
    – Joe Silverman
    Dec 17, 2013 at 16:41
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    $\begingroup$ @stankewicz There's no need to go to quadratic fields. Just take $E:y^2+xy=x^3-36/(j-1728)x-1/(j-1728)$ and substitute $j=1/p^N$ for some prime $p\ge5$. Since $j(E)=j$, its easy to check that $E$ has split multiplicative reduction at $p$ and $\text{ord}_p(\mathcal{D}_E)=N$. So that gives examples over $\mathbb{Q}$ with $c_p(E)$ arbitrarily large. $\endgroup$
    – Joe Silverman
    Dec 17, 2013 at 17:40
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    $\begingroup$ Actually there are many curves in the Cremona database whose Tamagawa numbers are divisible by $p\ge 11$, e.g. 147b2 ($p=13$), 190a1 ($p=11$), 262a1 ($p=11$) etc. There are $>6000$ of them of conductor up to 36000, and the claim is true for all of them, even without the "good ordinary at $p$" assumption, i.e. $p$ divides the degree of the modular parametrization in all these cases. So computationally the claim seems very feasible. $\endgroup$
    – Tim Dokchitser
    Dec 17, 2013 at 22:29
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    $\begingroup$ Maybe this argument should work: (I would post it as a comment but I am new here and I cannot) The newform on $\Gamma_0(N)$ associated to $E$ is $p$-isolated since $p$ does not divide the degree of the modular parametrisation. If $p$ divide a Tamagawa number, then you are in the setting of Ribet's lowering the level theorem. So the form should be congruent to form of lower level, contradicting the fact that $f$ is $p$-isolated. $\endgroup$
    – And85
    Aug 21, 2014 at 10:44

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