Why all supersingular elliptic curves over $\bar{\mathbb{F}_p}$ are isogenous?

Question

Lemma 3.2.1 in Baker, González-Jiménez, González, Poonen, "Finiteness theorems for modular curves of genus at least 2", Amer. J. Math. 127 (2005), 1325–1387.
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I don't understand why "all supersingular elliptic curves over $\bar{F_p}$ are isogenous". Could anyone help me?
This problem arises from Supersingular elliptic curves and their "functorial" structure over F_p^2.
Moreover, I found some similar question: Isogenies between elliptic curves and their endomorphism rings and Isogenies between supersingular elliptic curves.

Answer 1

This can be proved in several ways (this can be found in the literature as Lemma 42.1.11 in Voight's book on Quaternion algebras, or Proposition 5.2 in https://arxiv.org/pdf/2005.01537v1.pdf). Typically the proof relies on Tate's isogeny theorem (1966) which states, among other things, that if two elliptic curves over a finite field have the same number of rational points, then they are isogenous.

$$ \newcommand{\End}{\mathrm{End}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\C}{\mathbb{C}} $$

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Proposition. Let $p$ be a prime. Let $E,E'$ be two supersingular elliptic curves over $\F_{p^2}$. Then $E,E'$ are isogenous over $\F_{p^{24}}$.

(Note that the $j$-invariant of supersingular elliptic curves over $\overline{\F_p}$ belongs to $\F_{p^2}$, we may assume (up to $\overline{\F_p}$-isomorphism) that $E,E'$ are defined over $\F_{p^2}$).

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Proof.

Write $\alpha, \beta \in \C$ for the eigenvalues of the $p^2$-Frobenius map on the $\ell$-adic Tate module of $E$ for some $\ell \neq p$, so that $|E(\F_{p^{2r}})| = p^{2r} + 1 - (\alpha^r + \beta^r)$ for every $r \geq 1$. Since $E$ is supersingular, its trace $t_1 := \alpha + \beta$ is $\equiv 0 \pmod p$. By Hasse bound we have $|t_1| \leq 2 \sqrt{p^2} = 2p$, so we have $t_1 \in \{ -2p, -p, 0, p, 2p \}$.

In each case, $\alpha, \beta$ can be computed explicitly as roots of a quadratic polynomial (characteristic polynomial of Frobenius):

• If $t_1 = -2p$, then $X^2 + 2pX + p^2 = (X+p)^2 = 0$ yields $\alpha = \beta = -p$.

• If $t_1 = -p$, then $X^2 + pX + p^2 = 0$ yields $\alpha,\beta = \frac{-p + \sqrt{p^2 - 4p^2}}{2} = p \cdot \dfrac{-1 \pm i \sqrt{3}}{2}$

• If $t_1 = 0$, then $X^2 + p^2 = 0$ yields $\alpha = - \beta = ip$.

• If $t_1 = p$, then $X^2 - pX + p^2 = 0$ yields $\alpha,\beta = p \cdot \dfrac{1 \pm i \sqrt{3}}{2}$

• If $t_1 = 2p$, then $(X - p)^2 = 0$ yields $\alpha = \beta = p$.

In all cases, we observe that $\alpha^{12} = \beta^{12} = p^{12}$ (i.e. $\alpha,\beta$ are equal to $p$ times some $12$-th root of unity [in fact either of order $1,2,3,4$ or $6$]). This proves that $$ |E(\F_{p^{24}})| = p^{24} + 1 - (\alpha^{12} + \beta^{12}) = p^{24} + 1 - 2 \cdot p^{12} = |E'(\F_{p^{24}})| $$ so that $E,E'$ are isogenous over $\F_{p^{24}}$ by Tate's theorem. This concludes the proof. $\blacksquare$

Answer 2

Tate's isogeny theorem certainly suffices, but one can show much more. Consider the (multi)graph whose vertices are supersingular $j$-invariants mod $p$ and whose edges are isogenies of degree $2$. This family of graphs is not only connected, but it is Ramanujan [1, Prop. 3], meaning that its eigenvalue separation is as large as possible (the graph is as well-connected as possible). Therefore $2$-isogenies alone suffice to reach any supersingular curve from any other supersingular curve. The same statement holds for any other (prime) isogeny degree $\ell$ other than $p$. For example, $37$-isogenies would work (assuming $p \neq 37$).

The proof of the Ramanujan property is too long to repeat here, but essentially the largest eigenvalue of the $\ell$-isogeny graph is the $\ell$-th coefficient of the Eisenstein series $$ \sum_{n=1}^\infty \sigma_1(n) q^n = \sum_{n=1}^\infty \left(\sum_{d \mid n} d^1\right) q^n, $$ i.e. $\ell+1$ (which is also the degree of the graph), and the remaining eigenvalues are coefficients of cusp forms of weight $2$ on the modular curve $\Gamma_0(N)$ for some suitably chosen $N$, which are upper bounded by $\sigma_0(\ell) \sqrt{\ell} = 2\sqrt{\ell}$ by the Ramanujan-Petersson conjecture.

[1] Pizer, Ramanujan Graphs and Hecke Operators, Bulletin of the AMS 23 (1) 1990, pp. 127-137, https://doi.org/10.1090/S0273-0979-1990-15918-X