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 function fields of characteristic *p*. This problem was solved in 1994 by work of Michel Raynaud and David Harbater.

The problem involves a finite group*G*, a prime number *p*, and a nonsingular integral algebraic curve *C* defined over an algebraically closed field *K* of characteristic *p*.

The question addresses the existence of Galois extensions*L* of *K*(*C*), with *G* as Galois group, and with restricted ramification. From a geometric point of view *L* corresponds to another curve *C*′, and a morphism

Ramification geometrically, and by analogy with the case of Riemann surfaces, consists of a finite set*S* of points *x* on *C*, such that π restricted to the complement of *S* in *C* is an étale morphism. 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

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

This was proved by Raynaud.

For the general...

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The problem involves a finite group

The question addresses the existence of Galois extensions

- π :
*C*′ →*C*.

Ramification geometrically, and by analogy with the case of Riemann surfaces, consists of a finite set

The subgroup

*G*/*p*(*G*).

Then for the case of

*n*≤*s*.

This was proved by Raynaud.

For the general...

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