number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, the Katz–Lang finiteness theorem, proved by , states that if ''X'' is a smooth geometrically connected

scheme Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'', a BBC Scotland documentary TV series * The Scheme (band), an English pop band * ''The Scheme'', an action role-playing video game for the PC-8801, made by Quest Corporation * ...

of finite type over a

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'' that is finitely generated over the

prime field In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is wid ...

, and Ker(''X''/''K'') is the kernel of the maps between their abelianized

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

s, 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

* Theorems in number theory {{Numtheory-stub