Let $E$ be an elliptic curve over $\mathbb{Q}$. Then how to compute the $p$-torsion elements of $E$ over the $p$-adic field $\mathbb{Q}_p$ using SAGE or any other means ? At least can we say whether $E(\mathbb{Q}_p)[p]=0$ or not ?


Suppose $p>2$ and that $E$ has good reduction. If the reduction $\tilde E(\mathbb{F}_p)$ has no $p$-torsion then there is no $p$-torsion in $E(\mathbb{Q}_p)$. Otherwise look at the exact sequence $$ 0\to E(\mathbb{Q}_p)[p]\to \tilde E(\mathbb{F}_p)[p]\to \hat E(p\mathbb{Z}_p)/p\hat E(p\mathbb{Z}_p).$$ Here $\hat E$ is the formal group. So to check if a $p$-torsion point $\tilde P$ in the reduction lifts to a $p$-torsion point in $\mathbb{Q}_p$ do the following. Take any lift $P\in E(\mathbb{Q}_p)$. Then $Q=pP\in \hat E(p\mathbb{Z}_p)$ is the image of $\tilde P$ under the boundary map. Now check if $Q$ belongs to $p\hat E(p\mathbb{Z}_p) = \hat E(p^2\mathbb{Z}_p)$ by looking at the valuation at a parameter $t(Q)=-x(Q)/y(Q)$ of the formal group.

Refined versions of this will work for any type of reduction over any local field.

  • $\begingroup$ The Sage code given in the previous post (now deleted) was not working. Can you suggest some other code for computing the $p$-torsion over $\mathbb{Q}_p$ ? $\endgroup$ – Suman May 26 '16 at 10:07
  • $\begingroup$ Yes, it was working. And for small $p$ it is very good to use that code. Otherwise the above translates easily to code. But this site is not meant to answer that sort of questions. $\endgroup$ – Chris Wuthrich May 26 '16 at 10:13
  • $\begingroup$ I tried for $p=11$. May be that's why it was not working. $\endgroup$ – Suman May 27 '16 at 8:05
  • 2
    $\begingroup$ sage: E = EllipticCurve("57a1") sage: Ep = E.base_extend(Qp(11,100)) sage: f = Ep.torsion_polynomial(11) sage: r = f.roots() sage: len(r) 0 shows that there are no11-torsion on that curve over $\mathbb{Q}_{11}$. $\endgroup$ – Chris Wuthrich May 27 '16 at 9:17

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