The Fundamental Group of Affine Curves in Positive Characteristic
نویسنده
چکیده
It is shown that the commutator subgroup of the fundamental group of a smooth irreducible affine curve over an uncountable algebraically closed field k of positive characteristic is a profinite free group of rank equal to the cardinality of k.
منابع مشابه
A family of étale coverings of the affine line
This note was inspired by a colloquium talk given by S. S. Abhyankar at the Tata Institute, on the work of Abhyankar, Popp and Seiler (see [2]). It was pointed out in this talk that classical modular curves can be used to construct (by specialization) coverings of the affine line in positive characteristic. In this “modular” optic it seemed natural to consider Drinfel’d modular curves for const...
متن کاملFundamental Group in Nonzero Characteristic
A proof of freeness of the commutator subgroup of the fundamental group of a smooth irreducible affine curve over a countable algebraically closed field of nonzero characteristic. A description of the abelianizations of the fundamental groups of affine curves over an algebraically closed field of nonzero characteristic is also given.
متن کاملNormal Subgroups of the Algebraic Fundamental Group of Affine Curves in Positive Characteristic
Let π1(C) be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic p > 0 of countable cardinality. Let N be a normal (resp. characteristic) subgroup of π1(C). Under the hypothesis that the quotient π1(C)/N admits an infinitely generated Sylow p-subgroup, we prove that N is indeed isomorphic to a normal (resp. characteris...
متن کاملEfficient elliptic curve cryptosystems
Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation...
متن کاملAbsolute Anabelian Cuspidalizations of Proper Hyperbolic Curves
In this paper, we develop the theory of “cuspidalizations” of the étale fundamental group of a proper hyperbolic curve over a finite or nonarchimedean mixed-characteristic local field. The ultimate goal of this theory is the group-theoretic reconstruction of the étale fundamental group of an arbitrary open subscheme of the curve from the étale fundamental group of the full proper curve. We then...
متن کامل