On the structure of elliptic fields. II.

*(English. Russian original)*Zbl 0969.11022
Izv. Math. 62, No. 1, 1-18 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 1, 3-20 (1998).

For Part I, see Math. USSR, Izv. 27, 39-51 (1986; Zbl 0595.14022).

The paper is concerned with elliptic curves rather than elliptic function fields. The author considers the torsion groups of the elliptic curves \[ F:y^2=x^3+ rx+s\quad \text{and} \quad G:v^2= u^4+au^2+b \] over a cyclotomic field \(k\), of which the curve \(G\) can be carried over in \(F\) by a birational transformation. By Hasse’s theorem [see H. Hasse, Zur Theorie der abstrakten elliptischen Funktionenkörper I, II, III. J. Reine Angew. Math. 175, 55-62 (1936; Zbl 0014.14903); ibid., 69-88 (1936; Zbl 0014.24901); respectively ibid., 193-208 (1936; Zbl 0014.24902)] the torsion subgroup of order \(m\) on \(F\) over the algebraic closure \(\overline k\) of any field \(k\) of characteristic \(p\geq 0\) is \[ F_m(\overline k)=\mathbb{Z}/ m\mathbb{Z} \times \mathbb{Z}/m \mathbb{Z}\quad \text{if }p\nmid m. \] Assuming that \(m=p^t\), where \(p\) is a prime and \(t\) a positive integer, that \(k=\mathbb{Q} (\zeta)\) for \(\zeta\) a primitive \(m\)-th root of unity, and under the hypothesis that \(F_m(\overline k) \in \mathbb{Q}(\eta)\) for \(\eta\) a primitive \(p^\ell\)-th root of unity, the author “shows” that \(m\leq 5\). Moreover, he gives “explicitly” the coordinates of generators of \(F_m(\overline k)\) in \(\mathbb{Q}(\eta)\) for \(m=2,3,4,5\). A similar result is obtained for the curve \(G\).

The condition that \(F_m(\overline k) \in K\) for an arbitrary number field \(K\geq k\) and any positive integer \(m\) is known to imply that \(K\) contains the \(m\)-th roots of unity. Hence, in the above case, we have \(t\leq\ell\).

The reviewer was unable to verify the elementary, but gigantic computations combined in lemmas and propositions which the author needs in order to “prove” his results. This is true in particular since he refers back to earlier papers, among which is his proof of the boundedness conjecture in 1971. This “proof” is generally not believed. It was, however, proved by Merel [see L. Merel, Invent. Math. 124, 437-449 (1996; Zbl 0936.11037), and Oesterlé (unpublished)] that if there is a rational point on \(F\) of prime order \(p\) over a number field \(k\) of degree \(n=[k: \mathbb{Q}]\), then \(p\leq(1+3^{n\over 2})^2\). This, together with a theorem of Manin, according to which for a rational point on \(F\) over \(k\) of order \(p^\nu\) the exponent \(\nu\) is bounded (by a bound depending on \(k\) and \(p)\), establishes the boundedness conjecture in its strong form. In fact, the Manin bound was made explicit by Parent who showed that \(\nu\leq 129 (5^n-1)(3n)^6\). It should be mentioned in this connection that, e.g., Kubert [see D. S. Kubert, Proc. Lond. Math. Soc. (3) 33, 193-237 (1976; Zbl 0331.14010)] has benefitted from a lot of ideas of Dem’yanenko in proving a partial case of the boundedness conjecture.

The paper is concerned with elliptic curves rather than elliptic function fields. The author considers the torsion groups of the elliptic curves \[ F:y^2=x^3+ rx+s\quad \text{and} \quad G:v^2= u^4+au^2+b \] over a cyclotomic field \(k\), of which the curve \(G\) can be carried over in \(F\) by a birational transformation. By Hasse’s theorem [see H. Hasse, Zur Theorie der abstrakten elliptischen Funktionenkörper I, II, III. J. Reine Angew. Math. 175, 55-62 (1936; Zbl 0014.14903); ibid., 69-88 (1936; Zbl 0014.24901); respectively ibid., 193-208 (1936; Zbl 0014.24902)] the torsion subgroup of order \(m\) on \(F\) over the algebraic closure \(\overline k\) of any field \(k\) of characteristic \(p\geq 0\) is \[ F_m(\overline k)=\mathbb{Z}/ m\mathbb{Z} \times \mathbb{Z}/m \mathbb{Z}\quad \text{if }p\nmid m. \] Assuming that \(m=p^t\), where \(p\) is a prime and \(t\) a positive integer, that \(k=\mathbb{Q} (\zeta)\) for \(\zeta\) a primitive \(m\)-th root of unity, and under the hypothesis that \(F_m(\overline k) \in \mathbb{Q}(\eta)\) for \(\eta\) a primitive \(p^\ell\)-th root of unity, the author “shows” that \(m\leq 5\). Moreover, he gives “explicitly” the coordinates of generators of \(F_m(\overline k)\) in \(\mathbb{Q}(\eta)\) for \(m=2,3,4,5\). A similar result is obtained for the curve \(G\).

The condition that \(F_m(\overline k) \in K\) for an arbitrary number field \(K\geq k\) and any positive integer \(m\) is known to imply that \(K\) contains the \(m\)-th roots of unity. Hence, in the above case, we have \(t\leq\ell\).

The reviewer was unable to verify the elementary, but gigantic computations combined in lemmas and propositions which the author needs in order to “prove” his results. This is true in particular since he refers back to earlier papers, among which is his proof of the boundedness conjecture in 1971. This “proof” is generally not believed. It was, however, proved by Merel [see L. Merel, Invent. Math. 124, 437-449 (1996; Zbl 0936.11037), and Oesterlé (unpublished)] that if there is a rational point on \(F\) of prime order \(p\) over a number field \(k\) of degree \(n=[k: \mathbb{Q}]\), then \(p\leq(1+3^{n\over 2})^2\). This, together with a theorem of Manin, according to which for a rational point on \(F\) over \(k\) of order \(p^\nu\) the exponent \(\nu\) is bounded (by a bound depending on \(k\) and \(p)\), establishes the boundedness conjecture in its strong form. In fact, the Manin bound was made explicit by Parent who showed that \(\nu\leq 129 (5^n-1)(3n)^6\). It should be mentioned in this connection that, e.g., Kubert [see D. S. Kubert, Proc. Lond. Math. Soc. (3) 33, 193-237 (1976; Zbl 0331.14010)] has benefitted from a lot of ideas of Dem’yanenko in proving a partial case of the boundedness conjecture.

Reviewer: Horst G.Zimmer (Saarbrücken)