# Primes of bad reductions for quotients of elliptic curves

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 $$E$$.

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

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).