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Stickelberger elements and modular parametrizations of elliptic curves. (English) Zbl 0697.14023
The author begins a study of modular parametrizations of elliptic curves over \({\mathbb{Q}}\) which leads him to a basic conjecture that if E is a modular elliptic curve of level N, then there is a unique parametrization \(X_ 1(N)\to E\) for which the Manin constant is \(\pm 1\). In the course of his study the author naturally singles out, in each \({\mathbb{Q}}\)-isogeny class \({\mathcal A}\) (of modular elliptic curves of level N), an elliptic curve \(E_ 1\) whose \(X_ 1(N)\)-parametrization is optimal in the sense that if \(X_ 1(N)\to^{\pi}E\) is a parametrization of the curve \(E\in {\mathcal A}\) then there is an isogeny \(\beta: E_ 1\to E\) such that \[ \begin{matrix} X_1(N) & @>\pi>> & E_1 \\ &_\pi\searrow & \downarrow\text{\rlap{\(\gamma\)}} \\ && E \end{matrix} \text{\quad commutes.} \] commutes. It is then easy to see that, as a consequence of the above conjecture, the optimal curve in an isogeny class is the curve of minimal Parshin-Faltings height. Using results of K. Rubin [Invent. Math. 71, 339-364 (1983; Zbl 0513.14012)] the author is able to verify this equality for certain \({\mathbb{Q}}\)-isogeny classes of elliptic curves with complex multiplication.
The author also defines a collection \(\{\theta_ M:\) M a positive integer} of Stickelberger elements (with values in the \({\mathbb{Q}}\)-span of the lattices of Néron periods of E) which are closely related to the Stickelberger elements of B. Mazur and J. Tate [Duke Math. J. 54, 711-750 (1987; Zbl 0636.14004)]. By analogy with Stickelberger elements attached to totally real fields, the author conjectures certain integrality properties of \(\theta_ M\). He then shows that these integrality properties are implied by his basic conjecture on modular parametrizations. - Further investigation shows that the integrality of the p-adic measures (attached to \({\mathcal A})\) of Mazur and Swinnerton-Dyer is also implied by the author’s conjecture (at least for \(p\neq 2).\)
In addition, the author shows that if the basic conjecture is true for a class \({\mathcal A}\) then it is also true for all twists by quadratic characters which are unramified at the primes of additive reduction.
Finally, the author cites numerical verification of the basic conjecture for the 749 curves of conductor less than 200 listed in the Antwerp tables.
Reviewer: S.Kamienny

MSC:
14H45 Special algebraic curves and curves of low genus
14G25 Global ground fields in algebraic geometry
11F11 Holomorphic modular forms of integral weight
14H52 Elliptic curves
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