Timeline for What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?
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21 events
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| Dec 15, 2015 at 11:00 | vote | accept | The Thin Whistler | ||
| Dec 15, 2015 at 10:53 | vote | accept | The Thin Whistler | ||
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| Dec 15, 2015 at 10:47 | vote | accept | The Thin Whistler | ||
| Dec 15, 2015 at 10:53 | |||||
| Dec 15, 2015 at 10:43 | vote | accept | The Thin Whistler | ||
| Dec 15, 2015 at 10:47 | |||||
| Dec 15, 2015 at 10:43 | vote | accept | The Thin Whistler | ||
| Dec 15, 2015 at 10:43 | |||||
| Dec 15, 2015 at 10:37 | answer | added | jvo | timeline score: 3 | |
| Dec 15, 2015 at 10:08 | history | edited | The Thin Whistler | CC BY-SA 3.0 |
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| Dec 15, 2015 at 8:42 | answer | added | David Loeffler | timeline score: 6 | |
| Dec 15, 2015 at 8:11 | comment | added | The Thin Whistler | DavidLoeffler, @jvo: Thank you both for your helpful comments. David: Can you tell me to which paper of Kato's you refer? (Maybe send me a link?) | |
| Dec 15, 2015 at 4:00 | comment | added | tracing | $X_0(49)$ has additive, but potentially good, reduction at $7$, and doesn't acquire a good reduction Neron model over $\mathbb Q(\mu_{7^{\infty}}).$ | |
| Dec 14, 2015 at 19:19 | comment | added | jvo | @David Loeffler Yes, WLOG! Thanks for pointing this out, I was writing in haste without thinking. | |
| Dec 14, 2015 at 18:53 | comment | added | David Loeffler | @jvo Can't we easily arrange that k is p-power-free WLOG? Did you maybe mean "when k is coprime to p"? | |
| Dec 14, 2015 at 17:17 | history | edited | The Thin Whistler | CC BY-SA 3.0 |
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| Dec 14, 2015 at 17:11 | history | edited | The Thin Whistler | CC BY-SA 3.0 |
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| Dec 14, 2015 at 17:01 | comment | added | jvo | Adding to David Loeffler's comment: I do not know about the case of additive reduction at p for the the latter (which I believe is open), but for an elliptic curve E having good ordinary reduction at p, you might also find some relevant theorems in Darmon-Tian, "Heegner points over towers of Kummer extensions", Canad. J. Math. 62, No. 5 (2010) 1060-1082 (at least when k is p-power free). See Proposition 1.4 (assuming B+S-D), Theorem 1.8, or Theorem 1.9 for instance. Roughly, the arithmetic of E here exhibits similar behaviour to that of E in the Z_p^2 extension of an imaginary quadratic field. | |
| Dec 14, 2015 at 16:47 | comment | added | David Loeffler | Much is known over $\mathbf{Q}(\mu_{p^\infty})$. That the torsion is finite follows from work of Serre in the 1970s. It is also true that the rank is finite, but this is a very much deeper result (due to Kato in 2004). | |
| Dec 14, 2015 at 15:29 | history | edited | The Thin Whistler | CC BY-SA 3.0 |
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| Dec 14, 2015 at 14:44 | history | edited | The Thin Whistler |
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| Dec 14, 2015 at 13:56 | history | edited | The Thin Whistler | CC BY-SA 3.0 |
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| Dec 14, 2015 at 13:07 | history | edited | The Thin Whistler | CC BY-SA 3.0 |
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| Dec 14, 2015 at 13:02 | history | asked | The Thin Whistler | CC BY-SA 3.0 |