About: Abhyankar's conjecture

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In abstract algebra, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of algebraic function fields of characteristic p. The soluble case was solved by Serre in 1990 and the full conjecture was proved in 1994 by work of Michel Raynaud and David Harbater. The problem involves a finite group G, a prime number p, and the function field K(C) of a nonsingular integral algebraic curve C defined over an algebraically closed field K of characteristic p. π : C′ → C. G/p(G). n ≤ s. This was proved by Raynaud. n ≤ s + 2 g.

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rdfs:label	Abhyankar's conjecture (en)
rdfs:comment	In abstract algebra, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of algebraic function fields of characteristic p. The soluble case was solved by Serre in 1990 and the full conjecture was proved in 1994 by work of Michel Raynaud and David Harbater. The problem involves a finite group G, a prime number p, and the function field K(C) of a nonsingular integral algebraic curve C defined over an algebraically closed field K of characteristic p. π : C′ → C. G/p(G). n ≤ s. This was proved by Raynaud. n ≤ s + 2 g. (en)
differentFrom	Abhyankar's lemma
dcterms:subject	Algebraic curves Conjectures that have been proved Galois theory Theorems in abstract algebra
Wikipage page ID	3832555 (xsd:integer)
Wikipage revision ID	881960829 (xsd:integer)
Link from a Wikipage to another Wikipage	Purdue University David Harbater Algebraic function field Riemann surface Inverse Galois problem Genus (mathematics) Normal subgroup Morphism Conjecture Function field of an algebraic variety Galois extension Galois group Algebraic curve Field (mathematics) Projective line Prime number Algebraic curves Conjectures that have been proved Galois theory Theorems in abstract algebra Abstract algebra Characteristic (algebra) Sylow subgroup Algebraically closed Michel Raynaud Ramification (mathematics) Étale morphism Finite group Shreeram Abhyankar
Link from a Wikipage to an external page	http://www.math.purdue.edu/about/purview/spring95/conjecture.html
sameAs	Abhyankar's conjecture Abhyankar's conjecture Abhyankar's conjecture Abhyankar's conjecture
dbp:wikiPageUsesTemplate	dbt:Distinguish dbt:MathWorld dbt:Reflist
title	Abhyankar's conjecture (en)
urlname	AbhyankarsConjecture (en)
has abstract	In abstract algebra, Abhyankar's conjecture is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of algebraic function fields of characteristic p. The soluble case was solved by Serre in 1990 and the full conjecture was proved in 1994 by work of Michel Raynaud and David Harbater. The problem involves a finite group G, a prime number p, and the function field K(C) of a nonsingular integral algebraic curve C defined over an algebraically closed field K of characteristic p. The question addresses the existence of a Galois extension L of K(C), with G as Galois group, and with specified ramification. From a geometric point of view, L corresponds to another curve C′, together witha morphism π : C′ → C. Geometrically, the assertion that π is ramified at a finite set S of points on Cmeans that π restricted to the complement of S in C is an étale morphism.This is in analogy with the case of Riemann surfaces.In Abhyankar's conjecture, S is fixed, and the question is what G can be. This is therefore a special type of inverse Galois problem. The subgroup p(G) is defined to be the subgroup generated by all the Sylow subgroups of G for the prime number p. This is a normal subgroup, and the parameter n is defined as the minimum number of generators of G/p(G). Then for the case of C the projective line over K, the conjecture states that G can be realised as a Galois group of L, unramified outside S containing s + 1 points, if and only if n ≤ s. This was proved by Raynaud. For the general case, proved by Harbater, let g be the genus of C. Then G can be realised if and only if n ≤ s + 2 g. (en)
gold:hypernym	Conjecture
prov:wasDerivedFrom	wikipedia-en:Abhyankar's_conjecture?oldid=881960829&ns=0
page length (characters) of wiki page	3870 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Abhyankar's_conjecture
is Link from a Wikipage to another Wikipage of	List of conjectures David Harbater List of Marathi people in science, engineering and technology Timeline of Indian innovation Shreeram Shankar Abhyankar List of Indian Americans Abhyankar Abhyankar's lemma Cole Prize Michel Raynaud Abhyankar conjecture
is Wikipage redirect of	Abhyankar conjecture
is Wikipage disambiguates of	Abhyankar
is known for of	Shreeram Shankar Abhyankar
is known for of	David Harbater Shreeram Shankar Abhyankar Michel Raynaud
is foaf:primaryTopic of	wikipedia-en:Abhyankar's_conjecture

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