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On the Manin constants of modular elliptic curves. (English) Zbl 0749.14025
Arithmetic algebraic geometry, Proc. Conf., Texel/Neth. 1989, Prog. Math. 89, 25-39 (1991).
[For the entire collection see Zbl 0711.00011.]
Let $$E$$ be a strong modular curve of level $$M$$ and $$\Phi:X_ 0(M)\to E$$ be the corresponding strong modular parametrization, where $$X_ 0(M)_ \mathbb{Q}$$ is the modular curve over $$\mathbb{Q}$$ classifying elliptic curves with a given cyclic sugroup of order $$M$$ over $$\mathbb{Q}$$. Carayol has shown that the level $$M$$ of $$E$$ is equal to the conductor of $$E$$. Denote by $${\mathfrak E}$$ the Néron model of $$E$$ over $$\mathbb{Z}$$. Then, for a generating translation-invariant differential $$\omega$$ on $${\mathfrak E}$$, the differential form $$\Phi^*\omega$$ on $$X_ 0(M)_ \mathbb{Q}$$ is connected with the normalized newform $$\sum a_ nq^ n(dq/q)$$ associated with $$E$$ by the relation $$\Phi^*\omega=c\sum_{n\geq 1}q_ nq^ n(dq/q)$$, where the rational number $$c\in\mathbb{Q}^*$$ is the Manin constant. Manin conjectured that $$c=\pm 1$$.
The author takes an essential step towards proving Manin’s conjecture by showing that $$c$$ is an integer and that primes $$p>7$$ do not divide $$c$$ except possibly when $$E$$ has potentially ordinary reduction at $$p$$ of Kodaira type II, III, or IV in which case $$p$$ divides $$c$$ with multiplicity at most one. The reduction types $$I_ 0$$, $$I_ \nu$$, $$I^*_ 0$$ and $$I^*_ \nu$$ had been treated earlier by Mazur and Stevens, whereas the reduction types II, III, IV, $$\text{II}^*$$, $$\text{III}^*$$ and $$\text{IV}^*$$ are dealt with in the present paper. For some propositions, only sketches of proofs are given and for quite a few details, the reader is referred to two forthcoming papers of the author one of which is the version for publication of his 1989 thesis at Utrecht.

##### MSC:
 14H52 Elliptic curves 14G35 Modular and Shimura varieties 11F11 Holomorphic modular forms of integral weight
##### Keywords:
modular elliptic curves; Manin conjecture; Manin constant