Let $E$ be an elliptic curve over a number field $K$ and $p$ a prime. Suppose that $E$ has a $K$-rational $p$-torsion, which gives the short exact sequence $0\to\mathbb{Z}/p\to E[p]\to\mu_p\to0$ of Galois modules. Let us assume that this sequence splits, i.e. $E[p]\simeq\mathbb{Z}/p\times\mu_p$ as Galois modules. Set $E':=E/\mu_p$. Now, my questions is:

Is the set of bad primes for the elliptic curve $E'$ the same as the set of bad primes for $E$?

I am also wondering whether there is a general theory in this line, for example, how to compute the set of bad rimes for the quotient elliptic curve $E/H$ for "Galois stable" subgroup $H$ of $E$.

  • 3
    $\begingroup$ E and E’ are isogenous, so their places of bad reduction are the same by Neron Ogg Shafarevich $\endgroup$ Aug 17, 2023 at 15:33

1 Answer 1


As has already been remarked, but more generally, if $K$ is a number field and $A/K$ and $B/K$ are abelian varieties that are isogenous over $K$, then the criterion of Neron-Ogg-Shafarevich has as a quick corollary that $A$ and $B$ have the same set of primes of bad reduction. The paper with the proof is by Serre and Tate. (Serre, Jean-Pierre; Tate, J., Good reduction of abelian varieties, Ann. Math. (2) 88, 492-517 (1968). ZBL0172.46101.) If you want a proof just for elliptic curves, you could look at Section VII.7 in (Silverman, Joseph H., The arithmetic of elliptic curves, Graduate Texts in Mathematics 106. New York, NY: Springer (ISBN 978-0-387-09493-9/hbk; 978-0-387-09494-6/ebook). xx, 513 p. (2009). ZBL1194.11005).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.