criterion of Néron-Ogg-Shafarevich
Theorem (Criterion of Nron-Ogg-Shafarevich).
Let be an elliptic curve. Then has potential good reduction if and only if the inertia group acts on through a finite quotient for some prime .
Proof of Corollary.
() Assume that has potential good reduction. By definition, there exists a finite extension of , call it , such that has good reduction. We can extend (if necessary) so is a Galois finite extension.
Let and be the corresponding valuation and inertia group for . Then the theorem above ( (1)(3) ) implies that is unramified at for all , (since is a finite extension of ). So acts trivially on for all . Thus factors through the finite quotient .
() Let , and assume factors through a finite quotient, say . Let be the fixed field of , then is a finite extension, so we can find a finite extension so that . So the inertia group of is equal to , and acts trivially on . Hence the criterion ( (3)(1) ) implies that has good reduction over , and since is finite, has potential good reduction. ∎
() Assume , it is easy to prove that we can extend to a finite extension so that has a Weierstrass equation:
Since we are assuming , and:
then and ( ). Hence has good reduction, i.e. has potential good reduction.
() Assume that has potential good reduction, so there exists so that has good reduction. Let , the usual quantities associated to the Weierstrass equation over . Since has good reduction, , and so . But since is defined over , , so . ∎
|Title||criterion of Néron-Ogg-Shafarevich|
|Date of creation||2013-03-22 17:14:58|
|Last modified on||2013-03-22 17:14:58|
|Last modified by||alozano (2414)|
|Synonym||criterion of Neron-Ogg-Shafarevich|