$p$-Torsion

Fix

• elliptic curve $E$
• rigid prime $p$
• a prime $\ell\equiv 1\pmod{p}$ such that $\ell\nmid N_E$.

Let $K/\mathbb{Q}$ be the abelian extension corresponding to a character $\chi_{p,\ell}:(\mathbb{Z}/\ell\mathbb{Z})^* \rightarrow \boldsymbol{\mu}_p$ of order $p$ and conductor $\ell$.

The diagram we will plug into visibility theory is:

$$\xymatrix @=3pc{ {E[p]\,\,}\ar[r]\ar[d] & {E} \ar[dr]^{[p]}\ar[d] \\ {A}\ar[r] & {J} \ar[r]^{\tr} & {E.} }$$

Michael Stoll helped me to prove the following lemma.

Lemma 4.1   If $a_\ell(E)\not\equiv 2\pmod{p}$, then the following groups have no nontrivial $p$-torsion:

$$E(\mathbb{Q}_\ell), \quad J(\mathbb{Q}_\ell), \quad (J/E)(\mathbb{Q}_\ell), \quad \Phi_{A,\ell}(\mathbb{F}_\ell).$$

Proof.

• We first that prove
  $$J(\mathbb{Q}_\ell)[p]=0.$$
  By definition,
  $$J(\mathbb{Q}_\ell) = E_K(\mathbb{Q}_\ell\otimes _\mathbb{Q}K) \cong E(K_v)\times \cdots \times E(K_v),$$
  where $K_v$ is the completion of $K$ at the prime over $\ell$. The action of $\Frob_\ell\in\Gal(\mathbb{Q}_\ell^{\ur}/\mathbb{Q}_\ell)$ on
  $$E[p](\mathbb{Q}_{\ell}^{\text{ur}}) = E[p](\overline{\mathbb{Q}}_{\ell})$$
  has characteristic polynomial
  $$x^2 - a_\ell(E)x + \ell \in \mathbb{F}_p[x].$$
  This polynomial does not have $+1$ as a root, so
  $$E[p](\mathbb{Q}_{\ell}^{\text{ur}})=0.$$
  If $z\in E[p](K_v)$ then $\mathbb{Q}_\ell(z)\subset K_v$ and $K_v$ is totally ramified, so $\mathbb{Q}_{\ell}(z)=\mathbb{Q}_{\ell}$ and $z=0$. Thus $J(\mathbb{Q}_{\ell})[p]=0$.

• Next,
  $$(J/E)(\mathbb{Q}_{\ell}) \subset (J/E)(K_v) \approx E(K_v)\times \cdots \times E(K_v),$$
  so $(J/E)(\mathbb{Q}_{\ell})[p]=0$.

• Finally, consider $\Phi_{A,\ell}$. By Lang's Lemma,
  $$\mathcal{A}(\mathbb{F}_{\ell})\rightarrow\!\!\!\!\rightarrow \Phi_{A,\ell}(\mathbb{F}_{\ell}).$$
  Thus if $\Phi_{A,\ell}(\mathbb{F}_{\ell})[p]\neq 0$, then $\mathcal{A}(\mathbb{F}_{\ell})[p]\neq 0$. Since $p\neq \ell$, Hensel's lemma (and formal groups) imply that $A(\mathbb{Q}_{\ell})[p]\neq 0$, contrary to the fact that $J(\mathbb{Q}_{\ell})[p]=0$.

$\qedsymbol$