Timeline for Rational points techniques on curves not using their Jacobian
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 12, 2014 at 19:34 | answer | added | Michael Stoll | timeline score: 8 | |
| Aug 26, 2014 at 16:16 | comment | added | Maarten Derickx | The $C$ I have in mind are the modular curves $X^+_{ns}(p)$. If you can prove that these don't have rational points except the ones coming from CM for big enough $p$ you will have answered Serre's question whether for all non CM $E/\mathbb Q$ and all $l>37$ the $l$-adic galois representation associated to $E$ will be surjective. | |
| Aug 26, 2014 at 16:14 | comment | added | Maarten Derickx | You are right, there was a mistake, I indeed did can not proof density, only that the closure was of the same dimension, which still ruins Coleman-Chabauty. | |
| Aug 26, 2014 at 16:07 | history | edited | Maarten Derickx | CC BY-SA 3.0 |
added 25 characters in body
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| Aug 25, 2014 at 22:42 | comment | added | R.P. | I am wondering whether $J(\mathbb{Q})$ can indeed be dense in $J(\mathbb{Q}_p)$ for all $p$, as you say is true for your $C$. I actually think this can't ever happen, using an argument very much analogous to the one presented in this question: mathoverflow.net/questions/113968/… (to which you supplied the winning answer, incidentally). I am of course also curious what your $C$ is, if you'd care to disclose such information. :) | |
| Aug 25, 2014 at 19:42 | answer | added | Ariyan Javanpeykar | timeline score: 5 | |
| Aug 24, 2014 at 20:54 | comment | added | Damian Rössler | The very conjectural "effective Mordell conjecture" (see Astérisque 183) would provide such a method. The analog of this conjecture is proven over function fields (proofs by Arakelov and Szpiro) and such a method is thus available over function fields. Over number fields, I don't know any method that does not use the Jacobian. | |
| Aug 23, 2014 at 23:01 | history | asked | Maarten Derickx | CC BY-SA 3.0 |