Katz–Lang finiteness theorem

In number theory, the Katz–Lang finiteness theorem, proved by Nick Katz and Serge Lang (1981), states that if X is a smooth geometrically connected scheme of finite type over a field K that is finitely generated over the prime field, and Ker(X/K) is the kernel of the maps between their abelianized fundamental groups, then Ker(X/K) is finite if K has characteristic 0, and the part of the kernel coprime to p is finite if K has characteristic p > 0.

References

• Katz, Nicholas M.; Lang, Serge (1981), "Finiteness theorems in geometric classfield theory", L'Enseignement Mathématique, IIe Série, 27 (3), With an appendix by Kenneth A. Ribet: 285–319, doi:10.5169/seals-51753, ISSN 0013-8584, MR 0659153, Zbl 0495.14011