Mazur's Question on Mod $N$ Galois representations

In Rational Isogenies of Prime Degree, Mazur poses:

"the problem of determining all elliptic curves $E'/\mathbb{Q}$ with symplectic $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ isomorphisms $E'[N]\cong V$.

He then claims:

"This can be reduced to (or rephrased as) the problem of determining all $\mathbb{Q}$-rational points of a certain twisted form $X(V)$ of the modular curve $X(N)$."

My question is, how does one define $X(V)$ and how can one see that it is a twist of $X(N)$. Over which field is this twist defined?

• A. Silverberg ‘Explicit families of elliptic curves with prescribed mod N representations’, Modular forms and Fermat’s last theorem (eds G. Cornell, J. Silverman and G. Stevens; Springer, Berlin, 1997) 447–461. – Felipe Voloch Nov 30 '17 at 20:59

2 Answers

In addition to Felipe's reference, you can also have a look at Tom Fisher's papers https://www.dpmms.cam.ac.uk/~taf1000/papers/congr7and11.html and https://www.dpmms.cam.ac.uk/~taf1000/papers/congr9.html, where, in addition to defining these twists in general and giving lots of references, he works out explicit models for them for $N=7$, $9$, and $11$.

In addition to the previous comments, let me also say that this question has been considered by Kraus in joint works with Freitas, Halberstadt, and Oesterlé. The properties of the modular curve associated to the problem you mention are given in Proposition 1 of the paper by Kraus and Oesterlé, Sur une question de B. Mazur, Math. Ann. 293 (1992). The proof of this proposition is given in the reference [6] therein. Email me if you want an electronic copy of it.