Timeline for Infinite extensions such that every elliptic curve has finite rank
Current License: CC BY-SA 4.0
13 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| S Feb 4, 2020 at 5:01 | history | bounty ended | CommunityBot | ||
| S Feb 4, 2020 at 5:01 | history | notice removed | CommunityBot | ||
| Feb 2, 2020 at 18:34 | history | edited | R. van Dobben de Bruyn | CC BY-SA 4.0 |
Minor change in notation.
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| S Jan 27, 2020 at 3:49 | history | bounty started | R. van Dobben de Bruyn | ||
| S Jan 27, 2020 at 3:49 | history | notice added | R. van Dobben de Bruyn | Draw attention | |
| Jan 27, 2020 at 3:38 | history | edited | R. van Dobben de Bruyn | CC BY-SA 4.0 |
Highlighted main question.
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| Jan 25, 2018 at 7:35 | comment | added | Filippo Alberto Edoardo | For a nice survey in the Iwasawa theory setting, you can look at the introducion in Kim's paper in Journal of Number Theory 183 (2018) 352–387, expecially Theorem 1.2=5.17. | |
| Jan 22, 2018 at 23:11 | comment | added | Filippo Alberto Edoardo | Ah, I see. Still, as far as I understand, Greenberg proves boundedness of the $p$-part of $\mathrm{Sel}(E(F(\mathbb{Q}_n)))$ and to deduce boundedness of the Mordell–Weil one needs finiteness of Sha (at least the $p$-part) all over the tower... | |
| Jan 22, 2018 at 21:12 | comment | added | Álvaro Lozano-Robledo | For the Theorem that I quoted in my paper (and that you quote above), see Greenberg's article in the 1999 PCMI volume "Arithmetic Algebraic Geometry", Theorems 1.4 and 1.5. As far as I can tell the assumption that $E$ is defined over $\mathbb{Q}$ is essential because the theorems assume that the curve is modular. | |
| Jan 22, 2018 at 20:53 | comment | added | R. van Dobben de Bruyn | @FilippoAlbertoEdoardo: The Ribet attribution is probably for the finiteness of torsion; you need this on top of finite rank to deduce finite generation (although Rohrlich actually seems to prove finite generation). You might be right about the bad reduction actually; in Rohrlich's On $L$-functions of elliptic curves and cyclotomic towers, there seems to be an assumption on good reduction. | |
| Jan 22, 2018 at 19:41 | comment | added | Filippo Alberto Edoardo | In the quoted reference the theorem is vaguely attributed to Kato, Ribet (?) and Rohrlich. I am not sure whether this really holds over $\mathbb{Q}$, already. My feeling, for instance, is that if $p$ is a prime of bad reduction for $E$, little can be said. Since given any $E/\mathbb{Q}$ there exists one $p$ where $E$ has bad reduction, how can you show that $E(\mathbb{Q}(\mu_{p^\infty}))$ is finitely generated? | |
| Jan 22, 2018 at 19:21 | history | asked | R. van Dobben de Bruyn | CC BY-SA 3.0 |