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Hey all, I just ran into a question of Mazur from his 1978 paper 'Rational isogenies of Prime Degree' and I wonder what is the status of his question:

"Are there examples of elliptic curves $E/\mathbf{Q}$, $E'/\mathbf{Q}$ which are not isogenous over $\mathbf{Q}$ such that $E[N] \cong E'[N]$ (as $\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ modules) for some $N \geq 7$?"

Also the case $N=6$ is interesting, as it is the only genus $1$ case. For $N\leq 5$ the genus of $X(N)$ is zero, hence for all twists it will have infinitely many rational points.

Thanks for your answers.

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I want to add to the great answers below that there are examples with N = 17; see Fisher's paper on n-congruent elliptic curves mentioned by Stoll for one. These examples are interesting because the surface X_N is of general type for N >= 17. I believe it is unknown whether there are infinitely many non-trivial examples with N = 17. –  David Zureick-Brown Apr 18 '13 at 0:29
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3 Answers

up vote 13 down vote accepted

This question comes down to a question about a certain Hilbert modular surface X_N, which parametrizes pairs of elliptic curves with isomorphic N-torsion -- you can think of it as

X(N) x X(N) / PSL_2(Z/NZ)

where the group acts diagonally on the two factors.

This surface, like all Hilbert modular surfaces, has some modular curves lying around, which in this case parametrize pairs of elliptic curves that are isogenous. So you're asking: do these surfaces have rational points other than the ones lying on the special curves you know about? Well... questions about rational points on surfaces are hard. But for N = 7, the surface is rational so the points should be dense there. (Since only finitely many of the modular curves have non-cuspidal points over Q, this is enough!) When N = 11, the surface is elliptic. But for large N the surface becomes general type. Of course this still leaves us knowing basically nothing for certain about the presence or absence of rational points. Anyway, what I know about this all comes from David Carlton's thesis. As far as I know there's no substantial progress on the general question, which is one of my favorites!

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Dear Jordan, thanks a lot for stating the modular interpretation of the problem. –  Stefan Keil Feb 26 '13 at 8:44
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I once asked Imin Chen this very question around 13 years ago, and he pointed me to the 1992 paper "Sur une question de B. Mazur" by Kraus and Oesterlé. They work out an explicit example for $N = 7$. Imin also gave me examples for other $N \geq 8$:

$$\begin{aligned} N & = 11: & & \begin{aligned} E: & y^2 = x^3 + x^2 + 2 x + 2 \newline E': & y^2 + x y = x^3 + x^2 + 81710302x+ 576603336052 \end{aligned} \newline \newline N & = 11: & & \begin{aligned} E: & y^2 = x^3 - 3x - 34 \newline E': & y^2 = x^3 - 362988 x + 82933524 \end{aligned} \newline \newline N & = 11: & & \begin{aligned} E: & y^2 = x^3 - 27x + 918 \newline E': & y^2 = x^3 - 40332 x - 3071612 \end{aligned} \newline \newline N & = 13: & & \begin{aligned} E: & y^2 = x^3 + x - 10 \newline E': & y^2 = x^3 - 362249x + 165197113 \end{aligned} \end{aligned}$$

(I've never verified these claims myself.) Apparently the 1996 paper "Sur la comparaison galoisienne des points de torsion des courbes elliptiques" by Halberstadt and Kraus discusses what happens for composite $N$. As for what the Jordan Ellenberg mentioned, there is the 1998 paper "Modular diagonal quotient surfaces" by Kani and Schanz. (I didn't know David did this in his thesis. He was on my thesis defense committee, and we talked about mod 5 representations quite a bit!)

As for the other cases of $N \leq 6$, You may wish to take a look at the paper by Alice Silverberg entitled "Explicit families of elliptic curves with prescribed mod $N$ representations" in the volume "Modular forms and Fermat's last theorem". She works out formulas for $N = 3, 4, 5$. The papers "Mod 2 representations of elliptic curves" and "Mod 6 representations of elliptic curves" by Rubin and Silverberg round out the other cases.

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Dear Edray, thanks a lot for all the references and examples! –  Stefan Keil Feb 26 '13 at 8:43
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Tom Fisher has a paper where he shows that there are infinitely many examples for $N = 9$ and $N = 11$. See this link.

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Dear Michael, excellent reference, thanks a lot. –  Stefan Keil Feb 27 '13 at 13:06
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