Primes of the Form x2+ny2: Fermat, Class Field Theory, and Complex Multiplication

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 x² + ny². 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 = x² + ny². 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 ...