ank ar , ▁on ▁the ▁Gal ois ▁groups ▁of ▁algebraic ▁function ▁fields ▁of ▁characteristic ▁p . ▁The ▁sol ub le ▁case ▁was ▁solved ▁by ▁Ser re ▁in ▁ 1 9 9 0 ▁and ▁the ▁full ▁conject ure ▁was ▁proved ▁in ▁ 1 9 9 4 ▁by ▁work ▁of ▁Michel ▁Ray na ud ▁and ▁David ▁Har b ater . ▁ ▁The ▁problem ▁involves ▁a ▁finite ▁group ▁G , ▁a ▁prime ▁number ▁p , ▁and ▁the ▁function ▁field ▁K ( C ) ▁of ▁a ▁n ons ing ular ▁integral ▁algebraic ▁curve ▁C ▁defined ▁over ▁an ▁algebra ically ▁closed ▁field ▁K ▁of ▁characteristic ▁p . ▁ ▁The ▁question ▁addresses ▁the ▁existence ▁of ▁a ▁Gal ois ▁extension ▁L ▁of ▁K ( C ), ▁with ▁G ▁as ▁Gal ois ▁group , ▁and ▁with ▁specified ▁ram ification . ▁From ▁a ▁geometric ▁point ▁of ▁view , ▁L ▁corresponds ▁to ▁another ▁curve ▁C ′ , ▁together ▁with ▁a ▁morph ism ▁▁ π ▁: ▁C ′ ▁→ ▁C . ▁ ▁Ge omet r ically , ▁the ▁assertion ▁that ▁ π ▁is ▁ram ified ▁at ▁a ▁finite ▁set ▁S ▁of ▁points ▁on ▁C ▁means ▁that ▁ π ▁restricted ▁to ▁the ▁complement ▁of ▁S ▁in ▁C ▁is ▁an ▁ét ale ▁morph ism . ▁This ▁is ▁in ▁anal ogy ▁with ▁the ▁case ▁of ▁Riemann ▁surfaces . ▁In ▁Ab hy ank ar ' s ▁conject ure , ▁S ▁is ▁fixed , ▁and ▁the ▁question ▁is ▁what ▁G ▁can ▁be . ▁This ▁is ▁therefore ▁a ▁special ▁type ▁of ▁inverse ▁Gal ois ▁problem . ▁ ▁The ▁subgroup ▁p