mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Néron–Ogg–Shafarevich criterion states that if ''A'' is an

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...

over a

local field In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...

''K'' and ℓ is a

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

not dividing the characteristic of the

residue field In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and \mathfrak is a maximal ideal, then the residue field is the quotient ring k=R/\mathfrak, which is a field. Frequently, R is a local ri ...

of ''K'' then ''A'' has good reduction if and only if the ℓ-adic

Tate module In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group ''A''. Often, this construction is made in the following situation: ''G'' is a commutative group scheme over a field ''K'', '' ...

''T''_ℓ of ''A'' is unramified. introduced the criterion for elliptic curves. used the results of to extend it to abelian varieties, and named the criterion after Ogg, Néron and

Igor Shafarevich Igor Rostislavovich Shafarevich (; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. Outside mathematics, he wrote books and articles that criticised social ...

(commenting that Ogg's result seems to have been known to Shafarevich).

References

* * * Abelian varieties Elliptic curves Theorems in algebraic geometry Arithmetic geometry {{Algebraic-geometry-stub