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The global period $\Omega(E/K)$ of an elliptic curve defined over a number field $K$ is a product of local factors $\Omega_v(E_v/K_v)$, one for each infinite place $v$ of $K$. Here, $K_v$ denotes the completion of $K$ at $v$ (so $K_v=\R$ for a real place and $K_v=\C$ for a complex place), and $E_v$ denotes the base change of $E$ to $K_v$.

Fixing a Weierstrass model for $E$ with coefficients $a_i\in K$, a model for $E_v$ is given by the Weierstrass equation with coefficients $a_{i,v}$, the images of $a_i$ under $v$ in $K_v$. Associated to this model we have a discriminant $\Delta(E_v)$ and an invariant differential $\omega_v=dx/(2y+a_{1,v}x+a_{3,v})$.

For a real place given by an embedding $v:K\to\R$, we define $$\Omega_v(E_v) = \int_{E_v(\R)} \omega_E.$$ In terms of a basis of the period lattice of $E_v$ of the form $[x,yi]$ (when $\Delta(E_v)>0$) or $[2x,x+yi]$ (when $\Delta(E_v)<0$), where $x$ and $y$ are positive real numbers, we have $\Omega_v(E_v)=2x$.

For a complex place given by an embedding $v:K\to\C$, we define (note the factor of $2$) $$\Omega_v(E_v) = 2\int_{E_v(\C)} \omega_E\wedge\overline{\omega_E}.$$ In terms of a basis $[w_1,w_2]$ of the period lattice of $E_v$, where $\Im(w_2/w_1)>0$, we have $\Omega_v(E_v)=2\Im(\overline{w_1}w_2)$.

When $E$ has a global minimal model, we have $$\Omega(E/K) = \prod_{v}\Omega_v(E_v).$$ In general, given an arbitrary model for $E$ with discriminant $\Delta(E)$, we have $$\Omega(E/K) = \left|\frac{N(\Delta(E))}{N(\mathfrak{d}(E)}\right|^{1/12}\prod_{v}\Omega_v(E_v),$$ where $\mathfrak{d}$ is the minimal discriminant ideal of $E$ and $N(\mathfrak{d})$ denotes its norm.

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• Last edited by Andrew Sutherland on 2020-10-09 17:51:22
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