
doi: 10.1007/bf02434856
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.
ramification, characteristic \(p\), Elliptic curves, criterion of supersingularity of an elliptic curve, Singularities of curves, local rings, Ramification problems in algebraic geometry, Global ground fields in algebraic geometry
ramification, characteristic \(p\), Elliptic curves, criterion of supersingularity of an elliptic curve, Singularities of curves, local rings, Ramification problems in algebraic geometry, Global ground fields in algebraic geometry
| citations This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | 0 | |
| popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network. | Average | |
| influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | Average | |
| impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network. | Average |
