Timeline for Writing down minimal Weierstrass equations
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Jan 26, 2010 at 2:43 | comment | added | Emerton | Dear Qing, Do you know what Tate was worried about in the introduction of his paper on the algorithm, when he implies that some aspect of the algorithm is conjectural? | |
| Jan 25, 2010 at 23:54 | comment | added | Qing Liu | Tate's algorithm always ends because every step (except translations on $x$) decreases the discriminant of the equation. He works over algebraically closed residue field (perfect field would be OK but then we could have to work with quadratic or cubic extension of the residue field. The same remark holds over PID instead of DVR. | |
| Jan 4, 2010 at 4:20 | vote | accept | tkr | ||
| Jan 3, 2010 at 23:57 | comment | added | Emerton | I reread the introduction to Tate's article from Antwerp, and he remarks on various points being conjectural. But I'm not very sure of the details; perhaps someone else can clarify what he is referring to. | |
| Jan 3, 2010 at 23:40 | comment | added | tkr | I'm over a local field, not a number field so there is only one prime to consider. I will check out Tate's algorithm. | |
| Jan 3, 2010 at 23:14 | comment | added | David Zureick-Brown | My understanding is that the analogue of Tate's algorithm for higher genus hyperelliptic curves is not known to terminate, the problem being that one doesn't have a classification of regular proper minimal models of higher genus hyperelliptic curves for g > 2. I may be remembering incorrectly though.. | |
| Jan 3, 2010 at 23:12 | comment | added | David Zureick-Brown | Over a general number field there isn't a global minimal Weierstrauss model (this too is discussed in Silverman's AES I) and so one has to do it prime by prime. | |
| Jan 3, 2010 at 22:58 | comment | added | Anweshi | But the Tate algorithm is for just one prime. The PARI algorithm is perhaps a more heuristic one, considering all primes at once. | |
| Jan 3, 2010 at 22:05 | history | edited | Emerton | CC BY-SA 2.5 |
added 4 characters in body
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| Jan 3, 2010 at 21:00 | comment | added | JSE | Tate's algorithm is nicely explained in Silverman's second book, Advanced Topics in the Arithmetic of Elliptic Curves. It terminates as far as I know! But maybe there's a case I'm not thinking of. | |
| Jan 3, 2010 at 19:55 | history | answered | Emerton | CC BY-SA 2.5 |