Powered by OpenAIRE graph
Found an issue? Give us feedback
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Journal of Mathemati...arrow_drop_down
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
Journal of Mathematical Sciences
Article . 1997 . Peer-reviewed
License: Springer TDM
Data sources: Crossref
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
zbMATH Open
Article
Data sources: zbMATH Open
versions View all 2 versions
addClaim

This Research product is the result of merged Research products in OpenAIRE.

You have already added 0 works in your ORCID record related to the merged Research product.

A supersingularity criterion of elliptic curves

Authors: A. L. Smirnov;

A supersingularity criterion of elliptic curves

Abstract

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.

Keywords

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

  • BIP!
    Impact byBIP!
    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
Powered by OpenAIRE graph
Found an issue? Give us feedback
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).
BIP!Citations provided by BIP!
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.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
Average
Average
Upload OA version
Are you the author of this publication? Upload your Open Access version to Zenodo!
It’s fast and easy, just two clicks!