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I am in search of a concrete example [a concrete elliptic curve in Weierstrass form] of how Galois theory helps to find rational points on an elliptic curve. Chapter VI of Silverman and Tate discusses for instance the one-to-one homomorphism

$Gal(\mathbb{Q}(C[n])/\mathbb{Q})\to GL_2(\mathbb{Z}/n\mathbb{Z})$,

$C[n]$ denoting the points on the elliptic curve $C$ whose order divides $n$. It is discussed also that the field of definition of $C[n]$ is a Galois extension

$\mathbb{Q}(C[n]):\mathbb{Q}$,

etc. Can one extract a concrete help on finding rational points of $C$ out of this or other statements on Galois theory?

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  • $\begingroup$ Well, the size of the Galois group tells you roughly how many elements of C[n] are rational. That clearly helps. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 16, 2009 at 18:16

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What you've written down is relevant for finding rational torsion points on an elliptic curve. If that's what you want to do, Galois theory is certainly relevant. For instance, suppose you have an elliptic curve in Weierstrass form,

y^2 = f(x)

with f a cubic. Now suppose you find that f(x) has a linear factor (x-a). (I certainly take this to be a "Galois-theoretic" condition on f.) Then you've found a rational point of your curve, namely (a,0).

The relationship between Galois theory and points of infinite order is more subtle, involving Galois cohomology, and is discussed in chapter 10 of Silverman's book The Arithmetic of Elliptic Curves.

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  • $\begingroup$ Yes, this is what I want to do but I don't see how Galois theory 'helps' here to find rational points. In all the examples of Chapter VI, Silverman and Tate the points $C[n]$ are calculated by other methods before something is said about the Galois group. What I am searching for is a concrete elliptic curve in Weierstrass form $y^2 = f(x)$ such that one can determine $Gal((C[n]))$ 'easily' and gets then a conclusion about the rational points $\endgroup$
    – roger123
    Commented Dec 17, 2009 at 11:31
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In this example you can only be sure that there are NO rational points of order dividing n since the Galois group does not fix any n-torsion points. Certainly, you can always 'find rational points' of order n if you know the image of the homomorphism mentioned (at least, you can count them). You should just look which elements of $(\mathbb Z/n \mathbb Z)^2$ are fixed by this image.

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