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).

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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.

Reviewer: H.G.Zimmer (Saarbrücken)