$H^{0}$ cohomology group and elliptic curve

Question

Let $E$ be an elliptic curve with good reduction at $\ell$. Is it possible that one can have $H^{0}(\mathbb{Q}_{\ell}, E[\ell]) = E[\ell]$?

Answer 1

$\renewcommand{\Q}{\mathbb{Q}}$ This is only possible if $\ell = 2$. Indeed, for any field $K$, if there is an elliptic curve $E/K$ with full l-torsion -- $\# E(K)[\ell] = \ell^2$ -- then $K$ contains $\ell$ $\ell$th roots of unity (in particular has characteristic different from $\ell$). By standard algebraic number theory, the roots of unity in $\Q_{\ell}$ are $\pm 1$ for $\ell = 2$ and the $(\ell-1)$st roots of unity for each odd $\ell$.

The elliptic curve $E/\Q: y^2 + xy + y = x^3 + x^2 - 10x - 10$ has full $2$-torsion over $\mathbb{Q}$ and discriminant $(15)^4$ so has good reduction at $2$. It was obtained by looking in Cremona's Tables.

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Now let me revive some stuff from my deleted answer which I think is (more) interesting, though it perturbs the conditions of your question a bit.

By the above analysis, if an elliptic curve over a field $K$ of characteristic $0$ has full ${\ell}$-torsion, then $K \supset \Q(\mu_{\ell})$, the $\ell$th cyclotomic field. Merel and Stein analyzed the converse: when does there exist a curve over $\Q(\mu_{\ell})$ with full $\ell$-torsion? For very small $\ell$ there are some classically known examples -- e.g. the Fermat curve $X^3+Y^3+Z^3 = 0$ works for $\ell = 3$. There is also such a curve for $\ell = 5$. Merel and Stein (and Halberstadt for $\ell = 7$ and Rebolledo for $\ell = 13$) show that there are no such curves for any other $\ell < 1000$.

I claim that for any $p$-adic field $K$ and $n \in \mathbb{Z}^+$ the following are equivalent:
(i) $K$ contains the $n$th roots of unity.
(ii) There is an elliptic curve $E/K$ with full $n$-torsion.

This follows easily from the theory of Tate curves: if $E/\Q_{\ell}$ is any Tate curve with parameter $q$ which is a perfect $\ell$th power, then $E$ has full $\ell$-torsion iff $\Q_{\ell}$ contains the $\ell$th roots of unity. In fact this argument works with $\Q_{\ell}$ replaced by any complete discretely valued field of characteristic different from $\ell$.

The above result however does not mention good reduction. In fact, for a fixed positive integer $g$ and a fixed $p$-adic field $K$, $\sup \# A(K)[\operatorname{tors}]$ is finite -- indeed, explicitly bounded above -- as $A/K$ ranges over all abelian varieties of dimension $g$ with good reduction. In particular, for fixed $K$ and $g$, the set of primes $\ell$ such that $A$ can have a single point of order $\ell$, let alone full $\ell$-torsion, is finite. As above the set can be nonempty if $K$ contains the $\ell$th roots of unity: it always contains $\ell = 2$, as above.

Finally(?), there is a paper of Clark-Xarles which looks deeper at boundedness of torsion on abelian varieties over $p$-adic fields. Roughly speaking we prove that torsion is uniformly bounded as long as you stay away from abelian varieties which have a "p-adically uniformized piece": more precisely, as long as the Neron special fiber contains no nontrivial split torus. Our paper contains explicit bounds.