Timeline for Is there relationship between $\mu=0$ for an elliptic curve and the irreducibility of its residual representation at a prime $p$?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Dec 11, 2018 at 7:30 | vote | accept | CommunityBot | ||
| Nov 23, 2018 at 18:32 | comment | added | user130124 | Thank you for summarizing what's known over $\mathbb{Q}$, it was very helpful! | |
| Nov 23, 2018 at 18:28 | comment | added | user130124 | Yes the starting point would be to understand how to show that $\mu$ does not equal zero in some cases where the Galois representation is reducible over a number field and this can be a lot more subtle but perhaps doable. The elliptic curves should not be defined over $\mathbb{Q}$. I think one should only look at totally real fields to start with of course. | |
| Nov 23, 2018 at 10:13 | comment | added | Chris Wuthrich | If $E[p]$ is reducible then it can happen that all elliptic curves in the isogeny class have positive $\mu$-invariant. This is in "Iwasawa μ-invariants of elliptic curves and their symmetric powers" by Michael Drinen. So that part of Greenberg's conjecture for $\mathbb{Q}$ cannot hoped to be extended to arbitrary number fields. Whether it is still true that $\mu=0$ in the irreducible case, I don't know. | |
| Nov 23, 2018 at 9:41 | history | answered | Olivier | CC BY-SA 4.0 |