abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, Abhyankar's conjecture for affine curves is a

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

Shreeram Abhyankar Shreeram Shankar Abhyankar (22 July 1930 – 2 November 2012) was an Indian American mathematician known for his contributions to algebraic geometry. At the time of his death, he held the Marshall Distinguished Professor of Mathematics Chair ...

posed in 1957, on the

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

s of

algebraic function field In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebrai ...

s 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 Michel Raynaud (; 16 June 1938 – 10 March 2018 Décès de Michel Raynaud
So ...

and

David Harbater David Harbater (born December 19, 1952) is an American mathematician at the University of Pennsylvania, well known for his work in Galois theory, algebraic geometry and arithmetic geometry. Early life and education Harbater was born in New York ...

Statement

The problem involves a

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

''G'', a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

''p'', and the

function field Function field may refer to: *Function field of an algebraic variety *Function field (scheme theory) *Algebraic function field *Function field sieve *Function field analogy Function or functionality may refer to: Computing * Function key, a ty ...

''K(C)'' of a nonsingular integral

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

''C'' defined over an

algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra h ...

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''K'' of characteristic ''p''. The question addresses the existence of a

Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...

''L'' of ''K''(''C''), with ''G'' as Galois group, and with specified ramification. From a geometric point of view, ''L'' corresponds to another curve ', together with a

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

:π : ' → ''C''. Geometrically, the assertion that π is ramified at a finite set ''S'' of points on ''C'' means that π restricted to the complement of ''S'' in ''C'' is an

étale morphism In algebraic geometry, an étale morphism () is a morphism of Scheme (mathematics), schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topol ...

. This is in analogy with the case of

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

s. 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 In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved. There ...

Results

The subgroup ''p''(''G'') is defined to be the subgroup generated by all the

Sylow subgroup In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...

s of ''G'' for the prime number ''p''. This is a

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...

, and the parameter ''n'' is defined as the minimum number of generators of :''G''/''p''(''G''). Raynaud proved the case where ''C'' is the

projective line In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ...

over ''K'', the conjecture stating that ''G'' can be realised as a Galois group of ''L'', unramified outside ''S'' containing ''s'' + 1 points, if and only if :''n'' ≤ ''s''. The general case was proved by Harbater. Where ''g'' is the

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

of ''C'', ''G'' can be realised if and only if :''n'' ≤ ''s'' + 2 ''g''.

References

External links

* {{MathWorld, urlname=AbhyankarsConjecture, title=Abhyankar's conjecture
A layman's perspective of Abhyankar's conjecture
from

Purdue University Purdue University is a Public university#United States, public Land-grant university, land-grant research university in West Lafayette, Indiana, United States, and the flagship campus of the Purdue University system. The university was founded ...

Algebraic curves Galois theory Theorems in abstract algebra Conjectures that have been proved