Potential good reduction

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

See also

• Elliptic surface

References

• Serre, Jean-Pierre; Tate, John (1968). "Good reduction of abelian varieties". The Annals of Mathematics. Second 88 (3): 492–517. doi:10.2307/1970722.