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Trying to work out what the real period of an elliptic curve is as seen in the Birch Swinnerton-Dyer conjecture.

From what I've read, given an elliptic curve E over the rationals, one can associate to it a value $\displaystyle \Omega_{E} = \int_{E(\mathbb{R})}|\omega|$ where $\omega = \dfrac{dx}{2y + a_{1}x + a_{3}}$ as stated in

http://www.math.uci.edu/~asilverb/connectionstalk.pdf

I have also read that this is equal to twice the real period if the elliptic curve has two real components and just equal to the real period otherwise.

What is the real period of an elliptic curve and why is it well defined (that is if I have two isomorphic complex tori, why must they have the same real period)? Also is there a way to compute this integral?

Amzoti
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CAB
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2 Answers2

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The real period depends on the elliptic curve $E$ though of as a curve over the rationals, so it is not an invariant of $E$ thought of simply as a complex torus (i.e. it depends on $E_{/\mathbb Q}$, not just $E_{\mathbb C}$).

First of all, one has to normalize the choice of differential $\omega$; this is done by considering the Neron differential, i.e. $\omega$ is chosen so that it remains holomorphic on the Neron model of $E$. This pins down $\omega$ up to a sign (i.e. an element of $\mathbb Z^{\times}$, rather than just an element of $\mathbb Q^{\times}$).

Then you have to integrate $\omega$ over the real points $E(\mathbb R)$ of $E$, which form a closed curve on the Riemann surface $E(\mathbb C)$ of complex points of $E$. This gives the quantity denoted by $w_{\infty}$ in Wiles's write-up of the problem; the same quantity is denoted $\Omega_E$ in the wikipedia article.

In Tate's Inventiones article, this quanity is described as "either the positive real period of $\omega$ or twice that period, depending on whether $E(\mathbf R)$ is connected or has two components". So this suggests that the real period of $\omega$ means the integral of $\omega$ over the connected component of the identity in $E(\mathbb R)$. But in any case, it is the integral over all of $E(\mathbb R)$ that appears in the BSD formula.

Viktor Vaughn
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Matt E
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  • Okay so I see that I've overlooked something obvious to me now - curves isomorphic over $\mathbb{C}$ are not necessarily isomorphic over $\mathbb{Q}$. I guess of all the isomorphic curves to E, does choosing the minimal model give another way to guarantee that the associated lattice has the smallest real period? – CAB Jul 24 '13 at 03:34
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    @CAB, the curves $y^2=x^3+1$ and $y^2=x^3+2$ are isomorphic over $\mathbb{Q}(\sqrt[6]{2})$ but they are not isomorphic over $\mathbb{Q}$. – Álvaro Lozano-Robledo Jul 24 '13 at 18:07
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The real period can be computed very efficiently using the AGM. See my paper http://dx.doi.org/10.1016/j.jnt.2013.02.002

John Cremona
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