Timeline for How is the period of an elliptic curve defined exactly?
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 7, 2010 at 16:55 | vote | accept | CommunityBot | moved from User.Id=19475 by developer User.Id=69903 | |
| Jul 4, 2010 at 23:29 | answer | added | Chris Wuthrich | timeline score: 5 | |
| Jun 20, 2010 at 12:32 | comment | added | Robin Chapman | Wadim, here we go from the curve to the period. Given a nonzero holomorphic differential $\omega$ on an elliptic curve $E$ over $\mathbb{C}$ one gets a lattice $\Lambda$ in $\mathbb{C}$ by integrating $\omega$ over the elements of the homology group $H_1(E(\mathbb{C});\mathbb{Z})$. In the case of the Birch-Swinnerton-Dyer conjecture the curve $E$ is defined over $\mathbb{Q}$ and has a model $$y^2+a_1xy+a_3 y=x^3+a_2 x^2+a_4x+a_6$$ where the $a_i\in\mathbb{Z}$ minimize the discriminant. Here $\omega=dx/(2y+a_1 x+a_3)$ and the BSD formula features the smallest positive element of $\Lambda$. | |
| Jun 20, 2010 at 11:57 | comment | added | Wadim Zudilin | The answer depends on how you define an elliptic curve. An elliptic curve is a torus $\mathbb C/L$ where $L$ is a non-degenerate lattice. Any element of $L$ is a period. If $L=\mathbb Z\omega_1+\mathbb Z\omega_2$ for some $\Im(\omega_1/\omega_2)>0$, then one can get the elliptic curve in its "canonical" form $y^2=4x^3-g_2(L)x-g_3(L)$ using parametrisation $x=\wp(z;L)$ and $y=\wp'(z;L)$, by the Weierstrass function. | |
| Jun 20, 2010 at 10:53 | comment | added | Robin Chapman | The first one; also changing an orientation only changes the sign, and one wants the smallest positive real period. | |
| Jun 20, 2010 at 10:43 | history | asked | user19475 | CC BY-SA 2.5 |