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
[edit]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
[edit]For elliptic curves, potential good reduction is equivalent to the j-invariant being an algebraic integer.
See also
[edit]References
[edit]- Serre, Jean-Pierre; Tate, John (1968). "Good reduction of abelian varieties". The Annals of Mathematics. Second. 88 (3): 492–517. doi:10.2307/1970722. JSTOR 1970722. Zbl 0172.46101.