Timeline for What is the set of possible densities of pointless members in a family of rational curves over $\mathbb{Q}$?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Sep 29, 2013 at 13:50 | vote | accept | Vesselin Dimitrov | ||
| Sep 29, 2013 at 0:40 | comment | added | Felipe Voloch | Daniel has given a complete answer below. But for $x^2+y^2=n$ we can be more precise. This has a solution for $O(N/(\log N)^{1/2})$ of integers $n\le N$. I think this is due to Landau. | |
| Sep 29, 2013 at 0:14 | answer | added | Daniel Loughran | timeline score: 8 | |
| Sep 28, 2013 at 20:48 | comment | added | Vesselin Dimitrov | Actually no, I asked this question as a follow-up to #138581, which I only saw today. I have to admit that I do not even know the density of $q \in \mathbb{Q}$ for which $qy^2 = x^2 +1$ has a solution... (although this should be easy). I will think about this and other examples. But I was motivated by this basic observation: if we let $q$ run through the primes alone, then the density is $1/2$. | |
| Sep 28, 2013 at 20:22 | comment | added | Felipe Voloch | Do you have an example where $c>0$ but the family doesn't have a section? | |
| Sep 28, 2013 at 17:29 | history | edited | Vesselin Dimitrov | CC BY-SA 3.0 |
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| Sep 28, 2013 at 17:25 | comment | added | Vesselin Dimitrov | Thanks for spotting this, I will edit. I had considered the curves to be affine in relation to the other problem (in genus $> 1$, where the density should be $0$ unless there is a section); its purpose there was to remove a finite number of sections from a family of projective curves. But as you write, this has no significance in the genus zero case considered here. | |
| Sep 28, 2013 at 17:16 | comment | added | Jason Starr | One observation: you specify that the curves should be affine. I don't think that makes any difference. As soon as a (smooth, projective) rational curve has a single rational point, then it has a Zariski dense set of rational points. Thus, excluding finitely many points in the "boundary" will not change the problem. | |
| Sep 28, 2013 at 16:29 | history | edited | Vesselin Dimitrov | CC BY-SA 3.0 |
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| Sep 28, 2013 at 16:24 | history | edited | Vesselin Dimitrov | CC BY-SA 3.0 |
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| Sep 28, 2013 at 16:18 | history | asked | Vesselin Dimitrov | CC BY-SA 3.0 |