How is the period of an elliptic curve defined exactly? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T08:34:03Z http://mathoverflow.net/feeds/question/28829 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28829/how-is-the-period-of-an-elliptic-curve-defined-exactly How is the period of an elliptic curve defined exactly? norondion 2010-06-20T10:43:39Z 2010-07-04T23:29:38Z <p>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 $E(\mathbf{R})$.</p> <p>So what's the correct definition for the constant appearing in the BSD conjecture?</p> http://mathoverflow.net/questions/28829/how-is-the-period-of-an-elliptic-curve-defined-exactly/30568#30568 Answer by Chris Wuthrich for How is the period of an elliptic curve defined exactly? Chris Wuthrich 2010-07-04T23:29:38Z 2010-07-04T23:29:38Z <p>The comments above give already the answer, but for the sake of completeness let us be a bit more precise.</p> <p>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</p> <p>$$\frac{L^{*}(E,1)}{[E(\mathbb{R}):E(\mathbb{R})^{o}] \cdot \Omega^{+}} = \frac{\prod_{p} c_p \cdot | Sha| \cdot Reg}{| E(\mathbb{Q})_{tors}|^2}$$</p> <p>The denominator on the left, where the index is the number of connected components of $E(\mathbb{R})$, can also be written as the <i>absolute value</i> of $\int_{E(\mathbb{R})}\omega_E$ where $\omega_E$ is a invariant differential of a global minimal Weierstrass model.</p> <p>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 <a href="http://arxiv.org/abs/math/0610290" rel="nofollow"> Dokchitser's paper</a> 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 N&eacute;ron model.</p>