In mathematics, potential good reduction is a property of the reduction modulo a prime or, more generally,

prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...

, of an algebraic variety.

Definitions

Good reduction refers to the reduced variety having the same properties as the original, for example, an algebraic curve having the same genus, or a smooth variety remaining smooth. ''Potential'' good reduction refers to the situation over a sufficiently large finite extension of the field of definition.

Equivalent formulations

For elliptic curves, potential good reduction is equivalent to the j-invariant being an algebraic integer.

References

* {{cite journal , first1=Jean-Pierre , last1=Serre , authorlink1=Jean-Pierre Serre , first2=John , last2=Tate , authorlink2=John Tate (mathematician), title=Good reduction of abelian varieties , journal= The Annals of Mathematics , series=Second , volume=88 , number=3 , year=1968 , pages=492–517 , jstor=1970722 , zbl=0172.46101 , doi=10.2307/1970722 Abelian varieties

Definitions

Equivalent formulations

See also

References