In the first three chapters of the book, we solved our basic question concerning primes of the form *x*^{2} + *ny*^{2}. But the classical version of complex multiplication presented in Chapter Three does not do justice to more recent developments. In this final chapter of the book, we will discuss two additional topics, elliptic curves and Shimura reciprocity.

In the modern study of complex multiplication, elliptic functions are replaced with elliptic curves. In §14, we will give some of the basic definitions and theorems concerning elliptic curves, and we will discuss complex multiplication and elliptic curves over finite fields. Then, to illustrate the power of what we've done, we will examine two primality tests from the late 1980s that involve elliptic curves, one of which makes use of the class equation.

In §15, we turn our attention to a quite different topic, Shimura reciprocity. This concerns the deep interaction between Galois theory and special values of modular functions. We saw hints of this in §12 when we gave Weber's computation of *j*(). Using papers of Alice Gee and Peter Stevenhagen [A10, A11, A23] and Bumkyo Cho [A6] as a guide, we will revisit parts of §12 from this point of view and give an interesting twist on the question of *p* = *x*^{2} + *ny*^{2}.

The two sections of this chapter can be read independently of each other.

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