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.


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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