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**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.

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 |

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\textit{A. L. Smirnov}, J. Math. Sci., New York 83, No. 1, 158--160 (1994; Zbl 0880.14013); translation from Zap. Nauchn. Semin. POMI 211, 158--160 (1994)

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### 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 |

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