MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $S$ be an integral 1-dimensional scheme with function field $K$.

Let $E$ be an elliptic curve over $K$. The torsion of $E$ over $K$ is not necessarily finite. As an example consider an elliptic curve over $\mathbf C(t)$.

Now, assume the residue field of each closed point of $S$ to be finite.

Is the torsion of $E(K)$ finite?

Note that I'm not assuming $S$ to be noetherian.

Note that we may assume $S$ to be affine. To answer the above question positively, it would suffice to show the torsion embeds into the rational points of a special (not geometric) fibre of $E$.

share|cite|improve this question
up vote 5 down vote accepted

I have a vague idea that this might be wrong.

First, it seems to me that you can even assume that $S$ is the spectrum of a local ring. Then, if $p$ is the resicual characteristic, the reduction map will in general not be injective on $p$--torsion. So, if I had to produce couterexamples to this, I would start with an $E$ over $\mathbb Z_p$ with supersingular reduction, and then base change $E$ to some big, totally ramified extension of $\mathbb Z_p$ where lots of $p$--power torsion points of $E$ are defined.

share|cite|improve this answer
I think you can work backwards, choosing the subfield $K$ of an algebraic closure of $\mathbb{Q}_p$ by adjoining the coordinates of all $p$-power torsion points to $\mathbb{Q}_p$. – S. Carnahan Dec 7 '12 at 3:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.