×

A supersingularity criterion of elliptic curves. (English. Russian original) Zbl 0880.14013

J. Math. Sci., New York 83, No. 5, 662-663 (1997); translation from Zap. Nauchn. Semin. POMI 211, 158-160 (1994).
A well-known Belyi theorem states that an arbitrary algebraic curve defined over \(\overline \mathbb{Q}\) can be mapped onto the projective line \(\mathbb{P}^1\) so that the whole of ramification will be concentrated over three points of \(\mathbb{P}^1\) (we may assume that these points are \(\infty, 0,1)\). The converse is also true: A curve defined in characteristic zero and ramified over \(\mathbb{P}^1\) only at three points (with respect to the base) can be defined over an algebraic number field. In characteristic \(p>0\) the situation is different: Any curve, regardless of the transcendence degree of a definition field, can be ramified only over a single point of \(\mathbb{P}^1\). The following theorem is true.
Let \(k\) be a perfect field, \(\text{char} k =p>0\), let \(X\) be a complete smooth algebraic curve over \(k\). Then there exists a separable morphism \(\varphi: X\to \mathbb{P}^1\) defined over \(k\) and such that all ramification points of \(\varphi\) lie in \(\varphi^{-1}(\infty)\).
It is natural to try to describe geometric properties of algebraic curves in \(\mathbb{P}^1\), ramified only over \(\infty\). In the present paper, we give a criterion of supersingularity of an elliptic curve in terms of the morphisms of this curve into \(\mathbb{P}^1\) that are ramified only over a single point.

MSC:

14H20 Singularities of curves, local rings
14G25 Global ground fields in algebraic geometry
14H52 Elliptic curves
14E22 Ramification problems in algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Belyi, G. V., On Galois extensions of the maximal cyclotomic field, Izv. Akad. Nauk SSSR, Ser. Mat., 14, 247-256, (1980) · Zbl 0429.12004
[2] N. M. Katz, “Travaux de Laumon,” in:Sem. Bourbaki, 1987-88, No. 691, Asterisque (1988), pp. 105-132. · Zbl 0698.14014
[3] S. Lang,Elliptic Functions, Appendix 2 [in Russian], Moscow (1984). · Zbl 0551.14011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.