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I'm wondering if the following conjecture is true:

Let $\mathcal{A}$ be an isogeny class of elliptic curves over $\mathbf{Q}$. Fix an odd prime $p$ of good reduction. Then there is a curve $E \in \mathcal{A}$ such that the quantity $$\dfrac{L(E,1)}{\Omega_E}$$ is a $p$-adic unit.

That is, the quantity $\dfrac{L(E,1)}{\Omega_E}$ can always be made a $p$-adic unit after "shifting by an isogeny". I wanted to ask:

  1. Is this conjecture known to be true? It seems like if this is true, this would be well-known, but I'd like to confirm.

  2. If it is true, does anyone have a reference?

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    $\begingroup$ Firstly, this is wrong if the rank is positive. Then just take an isogeny class only containing one curve (the most frequent case) of rank 0 and look at your rational number. Chances are good that a good ordinary prime divides the numerator, because there is a Tamagawa number divisible by such a prime or the order of the Tate-Shafarevich group is. Somehow I would conjecture the opposite, that a good proportion of curves there is a prime that is a counter example to it. Of the first 7926 single rank 0 curves, 1215 fail your "conjecture". $\endgroup$
    – Chris Wuthrich
    Commented Dec 24, 2021 at 11:46
  • $\begingroup$ oops, that included $p=2$ as well. 1481 of all 16450 rank 0 isogeny classes of conductor < 10000 are counter-examples. The smallest is 114c with $p=5$. $\endgroup$
    – Chris Wuthrich
    Commented Dec 24, 2021 at 14:47

1 Answer 1

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The conjecture is False.

Let $\mathcal A$ be the isogeny class 6690j and $p=7$. The quantities $\dfrac{L(E,1)}{\Omega_E}$ for the two elliptic curves are $7$ and $49$ respectively, neither a $p$-adic unit.

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    $\begingroup$ Wonderful, thanks. Out of curiosity, how did you find that example? $\endgroup$
    – Adithya Chakravarthy
    Commented Dec 24, 2021 at 4:32
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    $\begingroup$ The point is to look for isogeny classes whose curves have large odd torsion. Actually there will always be a curve with no odd torsion; see this paper, so the point is to ensure that only one curve has no odd torsion and it has the quantity correct. $\endgroup$
    – LeechLattice
    Commented Dec 24, 2021 at 6:40

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