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In "An Introduction to the Theory of Numbers" by Hardy and Wright, they tantalizingly introduce a bunch of properties of elliptic curves, including the possibility of having Complex Multiplication, but they do not have time to show what it can be used for.

Can this extra structure somehow be used to help find rational points on an elliptic curve over Q?

Are there introductory level texts that describe how to use complex multiplication to solve Diophantine equations?

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  • $\begingroup$ There is some useful discussion of finding rational and integral points on elliptic curves in Smart, The Algorithmic Resolution of Diophantine Equations, but no mention of complex multiplication, which suggests to me that complex multiplication is not involved in this diophantine problem. But I'm out of my depth here. $\endgroup$ – Gerry Myerson Feb 2 at 0:36
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I recommend Advanced Topics in the Arithmetic of Elliptic Curves by Joseph Silverman. Chapter 2 is devoted to CM elliptic curves (or, if you only have access to his first book, there is some content in Appendix C.11).

I can't speak for finding $\mathbb{Q}$-rational points or solving Diophantine equations, but one particularly nice theorem give that $k(j(E)) = H$, the Hilbert class field of $k$ (the maximum unramified abelian extension of your base field $k$).

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