# CHAPTER THREE

# COMPLEX MULTIPLICATION

## §10. ELLIPTIC FUNCTIONS AND COMPLEX MULTIPLICATION

In Chapter Two we solved our problem of when a prime *p* can be written in the form *x*^{2} + *ny*^{2}. The criterion from Theorem 9.2 states that, with finitely many exceptions,

The key ingredient is the polynomial *f*_{n}(*x*), which we know is the minimal polynomial of a primitive element of the ring class field of []. But the proof of Theorem 9.2 doesn't explain how to find such a primitive element, so that currently we have only an abstract solution of the problem of *p* = *x*^{2} + *ny*^{2}. In this chapter, we will use modular functions and the theory of complex multiplication to give a systematic method for finding *f*_{n}(*x*).

In §10 we will study elliptic functions and introduce the idea of complex multiplication. A key role is played by the *j*-invariant of a lattice, and we will show that if is an order in an imaginary quadratic field *K*, then its *j*-invariant *j*() is an algebraic number. But before we can get to the real depth ...