# Abhyankar's conjecture

In abstract algebra, **Abhyankar's conjecture** is a 1957 conjecture of Shreeram Abhyankar, on the Galois groups of algebraic function fields of characteristic *p*.^{[1]} The soluble case was solved by Serre in 1990^{[2]} and the full conjecture was proved in 1994 by work of Michel Raynaud and David Harbater.^{[3]}^{[4]}^{[5]}

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 with
a morphism

- π :
*C*′ →*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.
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*.

## References

- ↑ Abhyankar, Shreeram (1957), "Coverings of Algebraic Curves",
*American Journal of Mathematics***79**(4): 825–856, doi:10.2307/2372438. - ↑ Serre, Jean-Pierre (1990), "Construction de revêtements étales de la droite affine en caractéristique p" (in French),
*Comptes Rendus de l'Académie des Sciences, Série I***311**(6): 341–346 - ↑ Raynaud, Michel (1994), "Revêtements de la droite affine en caractéristique p > 0",
*Inventiones Mathematicae***116**(1): 425–462, doi:10.1007/BF01231568, Bibcode: 1994InMat.116..425R. - ↑ Harbater, David (1994), "Abhyankar's conjecture on Galois groups over curves",
*Inventiones Mathematicae***117**(1): 1–25, doi:10.1007/BF01232232, Bibcode: 1994InMat.117....1H. - ↑ Fried, Michael D.; Jarden, Moshe (2008),
*Field arithmetic*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge,**11**(3rd ed.), Springer-Verlag, p. 70, ISBN 978-3-540-77269-9

## External links

- Weisstein, Eric W.. "Abhyankar's conjecture". http://mathworld.wolfram.com/AbhyankarsConjecture.html.
- A layman's perspective of Abhyankar's conjecture from Purdue University

Original source: https://en.wikipedia.org/wiki/Abhyankar's conjecture.
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