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Critical and ramification points of the modular parametrization of an elliptic curve. (English) Zbl 1082.11033

By the recent work of Wiles, Taylor, Breuil, Conrad and Diamond, every elliptic curve \(E\) over the field \(\mathbb{Q}\) of rational numbers is modular. Hence if \(E\) has conductor \(N\) there is a modular parametrization \(\varphi: X_0(N)\to E(\mathbb{C})\). Of course, \(E(\mathbb{C})\cong\mathbb{C}/\Lambda\) for a suitable lattice \(\Lambda\) in \(\mathbb{C}\) so that \(\varphi\) can be determined.
The topological degree \(d\in\mathbb{Z}\) can be computed but for finitely many points \(z\in\mathbb{C}/\Lambda\), \[ \#\{\varphi^{-1}(z)\}< d. \] These points \(z\) are called ramification points. They are images of the so-called critical points of \(X_0(N)\). There are \(2g-2\) or of them if \(g\) denotes the genus of the curve \(X_0(N)\).
Critical points are important since Mazur and Swinnerton-Dyer proved that the analytic rank of \(E\) over \(\mathbb{Q}\) is at most equal to the number of fundamental (in a certain sense) critical points of odd order and moreover, that these numbers have the same parity.
This statement was made more precise by the author. But the essential aim of this paper is the experimental determination of the critical points and the ramification points.
These points can theoretically be determined by a finite number of calculations, but in practice there is no efficient way of carrying out this task. We mention that critical points are the zeros of the differential \(d\varphi= 2\pi if(z)\,dz\), where \(f(z)\) is a newform of weight 2 on \(\Gamma_0(N)\).
More precisely, a point \(c\in X_0(N)\) is critical for \(\varphi\), if and only if \(c\) is a zero of \(f\) not counterbalanced by a pole of \(dz\). (It is interesting to observe that the zeros of \(f\) are often quadratic points of \(X_0(N)\).)
Thus the task is to localize the zeros of \(f(z)\). This can be done for involutory elliptic curves \(E\), i.e. curves for which there are operators \(u_1,u_2,\dots, u_k\) through which \(\varphi\) is completely factorizable: \[ \varphi: X_0(N)\to X_1\to X_2\to\cdots\to X_k\cong E(\mathbb{C}), \] where \(X_i= X_{i-1}/U_i\) and \(U_i\) is an involution of \(X_{i-1}\) \((i= 1,2,\dots, k)\).
The author takes examples from Cremona’s large tables. In his table 1 all involutory elliptic curves with conductor \(N\leq 100\) are listed.
Section 4 is devoted to the rank 2 elliptic curve \(E: y^2+ y= x^3+ x^2- 2x\). This curve appears to be prominent because it occurs already in N. P. Smart’s paper on \(S\)-integral points on elliptic curves [Math. Proc. Camb. Phil. Soc. 116, No. 3, 391–399 (1994; Zbl 0817.11031)] and in the paper of the underlined and his coauthors on the same topic [ibid. 127, 383–402 (1999; Zbl 0949.11033)].
The paper contains only a few misprints and not always the best possible notation is used, but altogether it is well-written.

MSC:

11G05 Elliptic curves over global fields
14H52 Elliptic curves
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14K15 Arithmetic ground fields for abelian varieties
11F11 Holomorphic modular forms of integral weight
11F25 Hecke-Petersson operators, differential operators (one variable)

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References:

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