Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Since every hyperelliptic curve $C$ of genus $g$ can be written as $y^2=f(x)$, is there an way to write an equation of a new curve $C'$ which has an etale cover over $C$ just using $x$ and $y$?

share|improve this question
add comment

1 Answer 1

Since nobody has answered this yet, here is my attempt, however Im not sure if it is exactly what you want. Hopefully it will at least lead you to a solution of your problem if you have not seen this kind of stuff already.

If your curve $C$ is defined over a field $k$ in which $f(x)$ splits as $f(x)=f_1(x)f_2(x)$, then we can define $C'$ via: $$C': y_1^2 = f_1(x) , y_2^2 = f(x).$$

Then the natural map $(x,y_1,y_2) \mapsto (x,y_1y_2)$ extends to an étale map on the compactifications.

However your question asks for $C'$ to "be written in terms of $x$ and $y$". If by this you mean present an affine patch of $C'$ which is a subset of $\mathbb{A}^2$, then $C'$ is at least birational to such an expression by the primitive element theorem and this shouldn't be too hard to write down - however it might be singular.

I hope this helps.

share|improve this answer
add comment

Your Answer

 
discard

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.