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Before asking my question, consider elliptic curves over complex numbers.

Let $E_{\tau}=\mathbb{C}/{\Lambda_{\tau}}$ be an elliptic curve over $\mathbb{C}$, where $\Lambda_\tau=\mathbb{Z} \oplus \tau \mathbb{Z}$ with $\text{Im} (\tau) > 0$.

Let $N>3$ be a prime number. Then, the number of cyclic subgroups of order $N$ of $E_\tau$ is $(N+1)$. They can be written of the form $C_i=\langle\frac{1+i\tau}{N} \rangle$ for $0 \leq i < N$ or $C_\infty = \langle \frac{\tau}{N} \rangle$.

Let $Y_0(N)$ be the (open) modular curve (over $\mathbb{C}$), which classifies the isomorphism classes of pairs $(E, C)$, where $E$ is an elliptic curve over $\mathbb{C}$ and $C$ is the cyclic subgroup of $E$ of order $N$. Then, $Y_0(N)(\mathbb{C})$ is isomorphic to $\mathbb{H}/\Gamma_0(N)$, where $\mathbb{H}$ denotes the complex upper half plane and $\Gamma_0(N)$ denotes the subgroup of upper triangular matrices modulo $N$ in $\text{SL}_2(\mathbb{Z})$. The points of $Y_0(N)$ can be written of the following form:

either $(E_{\tau}, C_0)$ with $\tau \in \mathbb{H}/{\Gamma_0(N)}$,

or $(E_\tau, C)$, where $C=C_i$ with $i=0, 1, \dots, N-1, \infty$ (and $\tau \in \mathbb{H}/{\text{SL}_2(\mathbb{Z})}$).

(There is a bijection between the above two.)

There is the Atkin-Lehner involution $w_N$ on $Y_0(N)$, which sends $(E, C)$ to $(E/C, E[N]/C)$, where $E[N]$ denotes the group of $N$-torsion points on $E$. (Note that $\tau \in \mathbb{H}/\Gamma_0(N)$ maps to $\frac{-1}{N\tau}$ by $w_N$.)

In general, there is no reason to have an isomorphism between $w_N(E, C)$ to $(E, C')$ for some $C'$ (with same $E$). However, that could happen in characteristic $p$. For instance, if $E$ is the only supersingular elliptic curve over $\overline{\mathbb{F}}_p$ (up to isomorphism) (e.g. $p=5$), then $E/C$ must be isomorphic to $E$ because $E/C$ is also supersingular.

Now, let $p$ be a prime number congruent to $-1$ modulo $3$, which is different from $N$. Let $\tau=e^{2\pi i/3}$ be a 3rd root of unity, and let $E:=E_\tau$. Then, the defining equation of $E$ is $y^2 = x^3+1$, which is supersingular over $\mathbb{F}_p$. So, $[(E, C_i)]$ can be regarded as a point of $Y_0(N)(\overline{\mathbb{F}}_{p})$, where $[(E, C_i)]$ denotes the isomorphism class of $(E, C_i)$. Note that $w_N$ also acts on $Y_0(N)(\overline{\mathbb{F}}_{p})$.

My question is as follows: is there an isomorphism $\varphi$ between $E$ and $E/{C_i}$ (over $\overline{\mathbb{F}}_p$)? If so, what is the inverse image of $E[N]/{C_i}$ by $\varphi$? (It must be a cyclic subgroup of order $N$ of $E$, so it is $C_j$ for some $j$.) That is, is this true: $$ w_N[(E, C_i)] = [(E, C_j)]? $$ If so, what's the relation between $i$ and $j$?

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  • $\begingroup$ I expect that if that happens, $i \cdot j \equiv 1 \pmod N$. $\endgroup$ – user1225 Aug 19 '17 at 4:28
  • $\begingroup$ The formula $i \cdot j=1$ mod $N$ cannot be correct because $E$ has an automorphism of order $6$ that sends $j$ to $-1/(j-1)$ so if any pair of $(i,j)$ work then $(i, -1/(j-1))$ must work as well. $\endgroup$ – Will Sawin Aug 22 '17 at 8:44
  • $\begingroup$ $w_N[(E,C_i)]=[(E,C_j)]$ if and only if there is a map $E \to E$ of degree $N$ whose kernel is $C_i$ and the kernel of whose transpose is $C_j$. The maps $E \to E$ form the maximal order of a quaternion algebra ramified at $p$ and $\infty$, the maps of degree $N$ are the elements of norm $N$, and you can see their kernels and cokernels using an isomorphism between the quaternion algebra and a matrix algebra over the $N$-adic completion. $\endgroup$ – Will Sawin Aug 22 '17 at 9:01
  • $\begingroup$ @Will; Let $i$ satisfy $i^2-i+1\equiv 0 \pmod N$, and $i \cdot j \equiv 1 \pmod N$. Then, $i$ and $j$ are both solutions of $x^2-x+1 \equiv 0 \pmod N$ and fixed by extra automorphism of $E$. Then, is this true that $w_N[(E, C_i)]=[(E, C_j)]? $\endgroup$ – user1225 Aug 22 '17 at 12:08
  • $\begingroup$ Yes, in fact that's true in characteristic zero, because then $(i \tau + 1, p ) $ is an ideal of index $p$ in $Z[\tau]$, which must have a single generator as that is a PID, and that generator gives an isomorphism between the lattice and $Z[\tau]$, hence an isomorphism between the quotient curve and $E$. $\endgroup$ – Will Sawin Aug 22 '17 at 12:23
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Even for $N=2$ it is not true that $E/C_i$ is necessarily isomorphic to $E$. In fact, the isogeny graph consisting of supersingular elliptic curves in char $p$ as vertices and $2$-isogenies as edges is connected, so as soon as there are more than one supersingular elliptic curve in char $p$, there will be an $i$ with $E/C_i$ not isomorphic to $E$.

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