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

Is it always possible to find an isogeny from a hyperelliptic curve of genus 4, to a 'normal' elliptic curve (genus 1), or a product of elliptic curves?

Are such isogenies easy to compute?

This question is motivated by a particular instance of a discrete logarithm problem. On hyperelliptic curves, computing group law in its jacobian is a slow operation. I'd like to convert the DLP to a group, where it's easier to solve. Any suggestions on different approaches are welcome too.

share|cite|improve this question
The Jacobian of a hyperelliptic curve can be simple, hence has no elliptic isogeny factor. – Qing Liu Sep 13 '11 at 8:10
up vote 5 down vote accepted

This is only a partial answer.

Let $C$ be a hyperelliptic curve and $E$ an elliptic curve. Let $i_1:C\to \mathbb{P}^1$ and $i_2: E\to\mathbb{P}^1$ be the double cover maps. Let $Q_1,\dots,Q_{2g+2}$ be the critical values of $i_1$ (i.e., the images of the Weierstrass points) and $P_1,\dots,P_4$ be the critical values of $i_2$.

Then there is an isogeny $C\to E$ if and only if there is a morphism $\varphi:\mathbb{P}^1\to\mathbb{P}^1$, such that the ramification indices at all the $Q_i$ are odd, at each other point of $\mathbb{P}^1$ the ramification indices are even and for all $i$ we have $\varphi(Q_i)\in \{P_1,\dots,P_2\}$. (This not so hard to prove, the only reference I know is a paper by Chad Schoen in Journal fuer Reine und Angewante Mathematik, you can use this to construct a lot of examples.)

For fixed $E$ and $g$ you can compute the dimension of the locus of hyperelliptic curves that admit an isogeny to $E$, at least over the complex numbers. Over the complex numbers this locus has dimension $g-1$ and therefore the locus of hyperelliptic curves admitting a morphism to an elliptic curve has dimension $g$, whereas the hyperelliptic locus has dimension $2g-1$. Hence a general complex hyperelliptic curve does not admit a morphism to an elliptic curve. I am quite sure a similar results holds true over finite fields, i.e., you need to calculate the dimension of a certain Hurwitz space of coverings $\mathbb{P}^1\to\mathbb{P}^1$. (Details of this calculation are in my paper on Noether-Lefschetz loci of elliptic surfaces, but I would not be suprised if someone had done this before I did this calculation.)

share|cite|improve this answer

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.