Timeline for Counting points on an algebraic set over a finite field
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 20, 2016 at 12:33 | history | edited | Sean Lawton | CC BY-SA 3.0 |
Minor edits.
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| Feb 20, 2012 at 17:31 | comment | added | Heinrich | Is there a simpler way to look at when $g=x+1/x$? Could you please explain? | |
| Feb 19, 2012 at 16:21 | comment | added | Felipe Voloch | Birch and Swinnerton-Dyer, Note on a problem of Chowla, Acta Arith, 5 (1959) 417-423, for value sets of general $g$. But for $x+1/x$ you won't need this. | |
| Feb 19, 2012 at 6:33 | comment | added | Heinrich | Thanks Felipe!It really helped. Can you please suggest me a reference to read on this? I was interested in the case when $g(x)=x+\frac{1}{x}$ and $f$ is any polynomial. | |
| Feb 18, 2012 at 21:44 | comment | added | Felipe Voloch | Expanding on Donu's comment, $C_g: y^p-y=f(g(x))$ maps to $C: y^p-y=f(x)$ by $(x,y) \mapsto (g(x),y)$. From general theory, it follows that the numerator of the zeta function of $C$ divides the numerator of the zeta function of $C_g$. One can also describe the number of points of $g(\mathbb{F}_q)$ in terms of the Galois group of $g(x)-t$, but the answer will depend a lot on $g$ and to get results about $C$ and $C_g$ you'll need to know whether $f$ and $g$ are related or not. Do you have a specific case that you are interested in? | |
| Feb 18, 2012 at 16:34 | comment | added | Heinrich | Yes, but I am looking for solutions over $g(\mathbb{F}_q)$ not over $\mathbb{F}_q$. Please let me know if its still unclear. Thanks! | |
| Feb 18, 2012 at 16:10 | comment | added | Donu Arapura | I don't really understand your notation. Is $C_g$ the curve $y^p-y=f(g(x))$? If not, then what? | |
| Feb 18, 2012 at 16:01 | history | edited | Heinrich | CC BY-SA 3.0 |
edited body
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| Feb 18, 2012 at 14:21 | history | edited | Heinrich |
edited tags
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| Feb 18, 2012 at 13:56 | history | asked | Heinrich | CC BY-SA 3.0 |