Timeline for Are there modular elliptic curves over a field extension of $\mathbb{Z}[i]$?
Current License: CC BY-SA 3.0
8 events
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| Oct 22, 2015 at 20:37 | comment | added | Joonas Ilmavirta | You should only post an answer if you have an answer; what you posted as an answer should have been an edit to the original question. I took the liberty of adding this text into the original question. Feel free to edit it. (It may help if you register your account.) | |
| Oct 22, 2015 at 20:35 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
Added an answer by the OP as an edit.
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| Apr 23, 2013 at 3:21 | comment | added | Pete L. Clark | @Robert: It sounds like you are looking for, among other things, the definition of the Hasse-Weil Zeta Function of a variety defined over a number field. Knowing the name will allow you to see that there is quite a large literature here. Also Felipe is right: based on what you write, you will need to learn some basic algebraic number theory in order to make sense of all this. (If you do not have this background, you can feel free to ask very basic questions on the related site math.stackexchange.com.) | |
| Apr 22, 2013 at 8:58 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Fixed formatting.
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| Apr 21, 2013 at 21:57 | comment | added | Keenan Kidwell | Just to echo François Brunault's comment, the ``modularity theorem" over, e.g., a totally real field, doesn't refer to the existence of a covering by a modular curve, but by the equality of the Hasse-Weil $L$-function with the $L$-function of a Hilbert modular eigenform of parallel weight $2$. Under certain hypotheses on the elliptic curve, this implies the existence of a covering by a Shimura curve, analogous to a modular curve, but these hypotheses do not always hold (they fail for example when the curve has good reduction everywhere and the field has even degree). | |
| Apr 21, 2013 at 21:31 | comment | added | François Brunault | What is your definition of modular? Do you mean "coming from modular/automorphic forms" or "quotient of a modular curve"? The two notions are not the same over an arbitrary number field -- the second one is more restrictive. | |
| Apr 21, 2013 at 18:12 | answer | added | Felipe Voloch | timeline score: 7 | |
| Apr 21, 2013 at 17:56 | history | asked | Robert Betts | CC BY-SA 3.0 |