# How is the period of an elliptic curve defined exactly?

I sometimes read $\int_{E(\mathbf{R})} \frac{dx}{2y + a_1x + a_3}$ and sometimes $\int_{E(\mathbf{R})} |\frac{dx}{2y + a_1x + a_3}|$. Furthermore, one has to choose an orientation on $E(\mathbf{R})$.

So what's the correct definition for the constant appearing in the BSD conjecture?

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The first one; also changing an orientation only changes the sign, and one wants the smallest positive real period. –  Robin Chapman Jun 20 '10 at 10:53
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. –  Wadim Zudilin Jun 20 '10 at 11:57
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 $\Lambda$. –  Robin Chapman Jun 20 '10 at 12:32
Thanks, Robin, for clarifying the OP. –  Wadim Zudilin Jun 20 '10 at 13:51

Let $E/\mathbb{Q}$ be an elliptic curve. Let $\Omega^{+}$ be the smallest positive element in the period lattice $\Lambda$. Then the conjecture of Birch and Swinnerton-Dyer predicts that
$$\frac{L^{*}(E,1)}{[E(\mathbb{R}):E(\mathbb{R})^{o}] \cdot \Omega^{+}} = \frac{\prod_{p} c_p \cdot | Sha| \cdot Reg}{| E(\mathbb{Q})_{tors}|^2}$$
The denominator on the left, where the index is the number of connected components of $E(\mathbb{R})$, can also be written as the absolute value of $\int_{E(\mathbb{R})}\omega_E$ where $\omega_E$ is a invariant differential of a global minimal Weierstrass model.
A better way of formulating the conjecture especially if $E$ is no longer defined over $\mathbb{Q}$ but over an arbitrary global field was given Tate. (See for instance conjecture 2.1 in Dokchitser's paper for a formulation). Since there are no global minimal models anymore one has to make either a conjecture that is invariant of the choice of a model or work with the Néron model.