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10

Theta Functions

Quotients of theta functions provide another source of elliptic functions. They are de-
fined for a lattice L of the form L τ = Zτ + Z with Im(τ) > 0. This is no restriction,
because every lattice L is equivalent to some such L τ . Since these functions f (z) are
always periodic in the sense f (z) = f (z + 1), we will consider their expansions in
∗
∗
∗
terms of q z = e 2πiz where f (z) = f (q z ) and f is defined on C = C −{0}.In §1
we consider various expansions in the variable q = q z of functions introduced in the
previous chapter.
Under the change of variable z to z + τ theta functions are not periodic but
periodic up to a specific factor. On the other hand they are holomorphic on the entire
plane. There are four specific theta functions which will give an embedding of a torus
into P 3 (C) such that the image of the torus is the locus of intersection of two quadric
surfaces in this three-dimensional space. This is another representation of an elliptic
curve as a curve in three space, see §8 of introduction.
An important feature of the theta-function picture of elliptic curves is that this
approach extends to the p-adic case while the Weierstrass definitions do not. This
was discovered by John Tate, and we give an introduction to Tate’s theory of p-adic
theta functions.
The basic reference for the first four sections is the first chapter of the book by
D. Mumford, Theta Functions I (Birkh¨ auser Boston).

§1. Jacobi q-Parametrization: Application to Real Curves

(1.1) Remark. For a lattice L with basis ω 1 and ω 2 with τ = ω 1 /ω 2 where Im(τ) >
0wehave L = ω 2 L τ where L τ = Zτ + Z. Now multiplication by ω 2 carries C/L
onto C/L τ isomorphically and substitution f (z) → f (ω 2 z) carries elliptic functions
for L isomorphically onto the field of elliptic functions for L τ . These considerations
were taken up in greater detail in 9(1.5) and 9(2.8).

In 11(1.4) we will determine how unique the invariant τ is among the lattices
L τ equal to a given L up to a nonzero complex scalar, but for now it suffices to