Timeline for Elliptic curves with Mordell-Weil group Z/2Z over Q
Current License: CC BY-SA 3.0
11 events
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| May 16, 2011 at 14:21 | comment | added | A. Pacetti | You can look at Knapp's book "elliptic curves", page 145 (although there are many other references) for families of elliptic curves with prescribed torsion (it does not work so good for torsion 2 since you can have torsion 4 given by Z/2 \times Z/2 or by Z/4, but the other cases works fine). | |
| May 15, 2011 at 16:30 | comment | added | Barinder Banwait | Using Cremona's database of the 'first' 910,603 curves (ordered by conductor), your density is 0.1757. It also seems to decrease as the conductor increases. | |
| May 13, 2011 at 18:39 | comment | added | Francesco Polizzi | @Franz: this is really interesting. Which orders? Could you please give some references? | |
| May 13, 2011 at 17:51 | comment | added | Franz Lemmermeyer | @Francesco: it's not really a matter of size. Families of elliptic curves whose torsion parts have subgroups of certain orders have been classified. | |
| May 13, 2011 at 11:00 | comment | added | Francesco Polizzi | Dear Franz and Minhyong, thank you for your comments. I'm actually not an expert in the field, so probably my idea of density is too vague and intuitive. What I really want to say in this answer in that elliptic curves with $E(\mathbb{Q})=\mathbb{Z}/2 \mathbb{Z}$ are probably "too many" to admith an exhaustive description. | |
| May 13, 2011 at 10:45 | comment | added | Minhyong Kim | That's right. If we consider elliptic curves as polynomials $x^3+Ax+B$ with integral $A,B$ ordered by size, you should get density zero for those with roots. From this point of view (and I think most other reasonable ones), the density of elliptic curves with 2-torsion should be zero. | |
| May 13, 2011 at 10:38 | comment | added | Franz Lemmermeyer | In addition, torsion of order 2 means that the defining polynomial on the right hand side splits into a linear and a quadratic factor over the rationals, plus the rather complicated condition of the absence of further torsion points of odd order. | |
| May 13, 2011 at 10:38 | comment | added | Franz Lemmermeyer | I don't think that the statement that curves with torsion groups of order 2 have density 1/2 is correct; it's probably, to paraphrase Pauli, not even false: these density statements only make sense if you think of elliptic curves as being ordered, and this can be done in several ways: with respect to the size of its coefficients, its discriminant, its conductor, etc. | |
| May 13, 2011 at 10:34 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| May 13, 2011 at 10:24 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| May 13, 2011 at 10:14 | history | answered | Francesco Polizzi | CC BY-SA 3.0 |