Timeline for Calculate reduction of Jacobian of hyperelliptic curve
Current License: CC BY-SA 3.0
12 events
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| Aug 18, 2014 at 10:47 | comment | added | Maxim | @jsm, if one use Liu algorithm (for defining stable reduction) directly it gives the Neron model possibly after extension of the base field. For me it isn't clear if the implemented algorithm gives the Neron model after extension of the base field or it gives the irreducible components over algebraic extension but for the Neron model of the given base field? | |
| Jul 16, 2014 at 9:51 | comment | added | jsm | For instance, if all the components happen to be defined over $\mathbb{F}_p$, then $\Phi(\mathbb{F}_p) =\Phi(\overline{\mathbb{F}_p})$, where $\Phi$ is the component group. | |
| Jul 16, 2014 at 9:47 | comment | added | jsm |
Magma doesn't compute the Neron model, only its group of components. You're right ComponentGroup only gives you the geometric component group. But Magma can also compute equations for the components using the intrinsic Components (which does not show up in the handbook). Then you can use the results of this paper by Bosch and Liu to find the $\mathbb{F}_p$-rational points of the component group using the $\mathrm{Gal}(\overline{\mathbb{F}_p}/\mathbb{F}_p)$-action on the components (Magma won't do it for you, though).
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| Jul 14, 2014 at 14:04 | comment | added | Maxim | @jsm, do you know, is it true that Magma's function calculate reduction and Neron model over algebraic closure of F_p? | |
| Jul 7, 2014 at 11:45 | vote | accept | Maxim | ||
| Jun 14, 2014 at 13:24 | comment | added | jsm | By the way, if your curve is hyperelliptic, then you can first call pMinimalWeierstrassModel to make sure that RegularModel starts with an equation that is minimal at 2. This won't affect the information you're interested in, but makes it much less likely that Magma can't compute a regular model. | |
| Jun 14, 2014 at 13:18 | comment | added | jsm | The curves $y^2 = f(x)$ and $y^2 = 16f(x)$ are isomorphic over $\mathbb{Q}$, so a regular model for one is isomorphic to a regular model for the other and vice versa. In any case, you should let the Magma group know about this. | |
| Jun 14, 2014 at 12:33 | comment | added | Maxim | @jsm, thanks. You are right. I've updated the example. It is interesting because there are two curves: $y^2 = f(x) = x^6 + 2*x^5 + x^4 + 2*x^3 + 6*x^2 + 4*x + 1$ and $y^2 = 16*f(x)$. In first case Magma will work as expected, but in second case -- there is error that I mentioned above. | |
| Jun 14, 2014 at 10:57 | comment | added | jsm | My version of Magma (V2.20-5) computes regular models at 2 for the examples you mention. If you don't have the latest version of Magma and you think you've found a bug, it's usually a good idea to first try the online-calculator. | |
| Jun 14, 2014 at 10:13 | comment | added | Maxim | I've updated the question and added examples that produce such error. Please look at them. | |
| Jun 14, 2014 at 6:52 | comment | added | Maxim |
Thanks a lot for your answer! It is almost the solution! But you are right there are exceptions in RegularModel function. For example in some cases it fails with error: Can't blow up fibre (failed to find an affine transformation to a line). So if you know some other way to deal with group of components it wold be great!
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| Jun 13, 2014 at 14:06 | history | answered | Michael Stoll | CC BY-SA 3.0 |