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Given an elliptic curve $E$ over $\mathbf{Q}$, we can attach two numbers two it.

  • the so-called Manin constant $c_E$. (Defined below the fold.)
  • the "algebraic $L$-value" given by $L(E,1)/\Omega_E$, where $\Omega_E$ is the period of $E$. This quotient is a rational number.

My question is: if $E$ is an optimal elliptic curve, what is the connection between these two numbers $c_E$ and $L(E,1)/\Omega_E$? Are they related in some way? In particular, if $c_E$ is a $p$-adic unit for some prime $p$, can one conclude that the $L$-value $L(E,1)/\Omega_E$ is also a $p$-adic unit? The reason I ask is because there are many theorems regarding the $p$-adic valuation of the Manin constant. I'd like to see if one can transfer those results to show things about the $p$-adic valuation of the $L$-values.

(Aside: Also, I am assuming that $E$ has analytic rank $0$ because otherwise the value $L(E,1)$ would be zero and the question wouldn't be interesting.)

Definition of Manin constant:

Since $E$ is modular, we have a modular parametrization $\phi: X_0(N) \to E$. So we can pull back the Neron differential $\omega_E$ on $E$ to get a differential form on $X_0(N)$: $$\phi^*(\omega_E) = c_E \, ( 2\pi i \, f(z) \, dz ) $$ where $f$ is the modular form attached to $E$. This defines the number $c_E$.

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    $\begingroup$ (Minor: you need to assume that you have chosen a modular parametrisation of minimal degree, then $c_E$ is defined up to $\pm1$.) Manin conjectured that the strong Weil curve (the optimal curve for $X_0$) has $c_E=1$. For most curves there is only one curve in the isogeny class and hence for those curves the conjecture is that $c_E=1$. Therefore you should not expect an implication in your direction. In the other direction, there are non-optimal curves with $c_E>1$ and this factor may then help to explain that the algebraic $L$-value is not a unit. $\endgroup$
    – Chris Wuthrich
    Commented Dec 17, 2021 at 16:26
  • $\begingroup$ @ChrisWuthrich I see. In that case, I'm wondering if there are any known facts about the p-adic valuation of the algebraic $L$-value? (This is just the $L$-value, nothing to do with the Manin constant.) For example, is it known that if $E$ is optimal, then $L(E,1)/\Omega_E$ is a $p$-adic unit? Perhaps we'd need to add conditions on the prime $p$ (maybe that $p \neq 2$ and $p$ is good ordinary). $\endgroup$
    – Adithya Chakravarthy
    Commented Dec 17, 2021 at 20:28
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    $\begingroup$ The $L$-value $L(E,1)$ can be 0. If it is nonzero, then the BSD conjecture predicts the $p$-adic valuation of the ratio $L(E,1) / \Omega_E^+$. Here is an example with $E=X_0(11)$ in PARI/GP: e = ellinit("11a1"); elllseries(e, 1)/e.omega[1] gives $0.2$, so it is not a $5$-adic unit. $\endgroup$
    – François Brunault
    Commented Dec 18, 2021 at 4:12

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