Period lattice of an elliptic curve (awaiting review)

For $E$ an elliptic curve defined over $\C$ by a Weierstrass equation with coefficients $a_1,a_2,a_3,a_4,a_6$, the period lattice of $E$ is the set $\Lambda$ of periods of the invariant differential $dx/(2y+a_1x+a_3)$, which is a discrete lattice of rank $2$ in $\C$. There is an isomorphism (of complex Lie groups) $\C/\Lambda \cong E(\C)$ defined in terms of the Weierstrass $\wp$-function.

For elliptic curves defined over $\R$ (and in particular, for those defined over $\Q$), the period lattice has one of two possible types dependng on the sign if the discriminant $\Delta$ of $E$:

• If $\Delta>0$, then $\Lambda$ is rectangular, with a $\Z$-basis of the form $\left<x,yi\right>$, where $x$ and $y$ are positive real numbers; in this case, $E(\R)$ has two connected components.
• If $\Delta<0$, then $\Lambda$ has a $\Z$-basis of the form $\left<2x,x+yi\right>$, where $x$ and $y$ are positive real numbers; in this case, $E(\R)$ has one connected component.

The real period of $E$ is defined to be $2x$ in each case, so is equal to the smallest positive real period multiplied by the number of real components.

Note that the period lattice depends on the choice of Weierstrass model of $E$; different models have homothetic lattices. For elliptic curves defined over $\Q$, the period lattice associated to a global minimal model of $E$ is called the Néron lattice of $E$. The real period of the Néron lattice is denoted $\Omega_E$, and appears in the Birch Swinnerton-Dyer conjecture for $E$.