Timeline for Rational points on the curve y^p=f(x) in characteristic p
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 11, 2013 at 11:48 | comment | added | Alex | I think I know how to prove this. Thanks for the helpful discussion. | |
| Jun 11, 2013 at 11:37 | comment | added | Felipe Voloch | I think you are right. The existence of a change of variables over a separable extension should be equivalent to your family of curves having infinitely many points over a separable extension. I don't know the answer to the question in your first comment. | |
| Jun 11, 2013 at 8:38 | comment | added | Alex | This doesn't seem to be correct as the example $f=tx^2+1$ over $K=\mathbb{F}_q(t)$ ($p>2$) shows. In this example the required change of variables is defined over a quadratic extension. Possibly the existence of such a change of variables over some separable extension is necessary, but it is not clear whether it is sufficient. | |
| Jun 10, 2013 at 20:28 | comment | added | Felipe Voloch | My guess would be that it's only for those polynomials for which there is a change of variables over $K$ taking the equation to one defined over $\mathbb{F}_q$. | |
| Jun 10, 2013 at 19:57 | comment | added | Alex | Thanks, this is helpful. Perhaps you can also answer the following question: is there a good characterization of all polynomials $f\in K[x]$ s.t. $y^{p^k}=f(x)$ has infinitely many K-points for all k? | |
| Jun 10, 2013 at 19:48 | vote | accept | Alex | ||
| Jun 10, 2013 at 13:16 | history | answered | Felipe Voloch | CC BY-SA 3.0 |