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I recall that the Manin constant for a strong elliptic curve is a rational integer $c_E$ such that, for a modular parametrization $\phi: X_1(N) \to E$, one has $\phi^*(\omega_E)= 2\pi i c_E f(z)\mathrm{d}z$ ($f$ is the modular form associated to $E$). The Manin constant is supposed to equal +/-1, but it is not proved yet.

I know that the prime divisors of $c_E$ are well known, but is there any bound for $c_E$, as the conducteur varies? If any, is there any reference for a proof? I heard that Edixhoven proved that $c_E$ is bounded (independently of $N$), but the article is forthcoming...

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up vote 16 down vote accepted

A useful reference on this topic (which maybe you know?) is "The Manin Constant", by Agashe, Ribet, and Stein, available here. On p. 3 they write

B. Edixhoven also has unpublished results (see [Edi89]) which assert that the only primes that can divide $c_E$ are 2, 3, 5, and 7; he also gives bounds that are independent of $E$ on the valuations of $c_E$ at 2, 3, 5, and 7. His arguments rely on the construction of certain stable integral models for $X_0(p^2 )$.

The reference [Edi89] is to Edixhoven's thesis, which is unpublished, but which can be found here at Edixhoven's website. At the end of the introduction to his thesis, Edixhoven writes

Finally, in the last section, we derive some results concerning the constant "c" attached to a strong Weil curve $E$. Manin conjectured that $c=1$. It is known that $c$ is a positive integer and Mazur proved that only 2 and primes where $E$ has additive reduction can divide $c$. Our methods show that primes $p>7$ where $E$ has additive reduction divide $c$ at most once, and in fact, for most of the possible reduction types (=Kodaira symbols), not (Theorem 4.6.3 and the remarks following this theorem). It might well be that a computation involving the period lattices of normalized newforms can solve the problem in the case of potentially good, ordinary reduction of type $II, III$ or $IV$. It should also be tried to get bounds on the exponents of 2, 3, 5 and 7 in $c$.

This suggests that the statement of Agashe, Ribet, and Stein is perhaps a little strong, in that Edixhoven does not give uniform bounds at all primes, but only at primes $p > 7$, with the suggestion that one could also hope to obtain such bounds at 2, 3, 5, and 7. You might try writing to Edixhoven for clarification, or perhaps also to one of Agashe, Ribet, or Stein to find out more precisely what they had in mind.

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Your answer is all the more helpful, since the reference you quote learns to me that $p|c_E$ implies $p^2|4N$, and $4|c_E$ implies $4|N$. In the case I study, $E$ is given by $y^2=x(x−a)(x+b)$ where $a+b+c=0$ and $a, b, c$ are coprime. Thus $N$ is quadrafrei and $c_E$ is at most 2. Thanks (and I guess I should write to one of the authors for more details about the problem you raise)! –  Bernikov Aug 28 '10 at 18:38
    
Edixhoven has been more or less claiming these uniform bounds (the ones mentioned by ARS) for some time. I think I saw him in 1997, then again in 2003, and he's always too busy. –  Junkie Feb 28 '11 at 12:34
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