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The Manin constant is defined for elliptic curves over $\Q$ which are optimal. Let $E$ be an optimal elliptic curve of conductor $N$, let $f$ be the modular form associated to $E$, and let $\varphi:X_0(N)\to E$ be the associated modular parametrization. Let $\omega_E$ be the Néron differential on $E$. Then the pull-back $\varphi^*\omega_E$ of $\omega_E$ to $X_0(N)$ satisfies \[ \varphi^*\omega_E = c \cdot 2\pi i f(z)dz \] for some non-zero rational number $c$, called the Manin constant of $E$.

It is conjectured that $c=1$ always. A theorem of Edixhoven states that $c\in\Z$, and there are several results stating that $c=1$ if certain conditions hold: see Amod Agashe, Ken Ribet and William Stein: The Manin Constant, Pure and Applied Mathematics Quarterly, Vol. 2 no.2 (2006), pp. 617–636. In an appendix to that paper, John Cremona gives an algorithm for verifying that $c=1$ in individual cases, and proves that $c=1$ for all optimal elliptic curves over $\Q$ in the database. Kęstutis Česnavičius proves $c=1$ for semistable elliptic curves over $\Q$, and more generally that $v_p(c) = 0$ if $p^2 \nmid N$ in The Manin constant in the semistable case, Compositio Math. 154 (2018), 1889–1920.

For non-optimal elliptic curves $E'$ over $\Q$, the Manin constant may also be defined, in terms of the Manin constant of the unique optimal curve isogenous to $E'$. Let $\varphi:X_0(N)\to E$ and $f$ be as above, and $\psi:E\to E'$ an isogeny of least degree from $E$ to $E'$. Then we obtain a parametrization $\psi\circ\varphi:X_0(N)\to E'$ and define the Manin constant $c'$ of $E'$ to be the non-zero rational number such that \[ (\psi\circ\varphi)^*\omega_{E'} = c' \cdot 2\pi i f(z)dz. \] This is an integer multiple of the Manin constant of $E$, since $\psi^*\omega_{E'}$ is an integer multiple of $\omega_E$; the multiplier divides the degree of $\psi$ but may be strictly less: it may equal $1$.

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  • Review status: reviewed
  • Last edited by Timo Keller on 2022-07-12 04:15:13
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