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There is a book about rational points on elliptic curves. What about algebraic points?

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What about them? Silverman has written other books on elliptic curves, e.g. The Arithmetic of Elliptic Curves and Advanced Topics in the Arithmetic of Elliptic Curves. What do you want to know? –  Qiaochu Yuan Jan 17 '11 at 20:44
    
Qiaochu Yuan, are they important to study and if so what is their basic theory. –  quanta Jan 17 '11 at 21:21
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It is quite common to study elliptic curves over $K$ or to ask about all $K$-points on an elliptic curve (or on an abelian variety), where $K$ is a subfield of the algebraic numbers; usually a number field (finite extension of $\mathbb{Q}$), but sometimes somewhat bigger (e.g., the extension given by adding all $p^n$th roots of unity for some fixed prime $p$). It is also common to work over finite fields, or over general global fields or over local fields. Most of the basic theory over number fields is very similar to that over $\mathbb{Q}$; e.g., the Mordell-Weil Theorem holds for any number field (the group of $K$-points on an elliptic curve defined over $K$ is finitely generated for any number field $K$). It's known that for any number field $K$, there is a finite list of possible torsion subgroups of the group of $K$-defined points of any elliptic curve defined over $K$ (that is, there is a version of Mazur's Theorem for each number field). There is also the version of integral points ($\mathcal{O}_K$-points, where $\mathcal{O}_K$ is the group of integers of $K$), etc.

In fact, most of Silverman's books mentioned by Qiaochu work over arbitrary fields, restricting on occasion to local fields or to global fields. Most of the theory is quite general, and yes, there is interest in understanding elliptic curves over number fields.

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