## 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
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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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