1834 CHAPTER 58. ELLIPTIC FUNCTIONS

Definition 58.1.17 When an analytic or meromorphic function is invariant under a groupof linear transformations, it is called an automorphic function. A function which is auto-morphic with respect to a subgroup of the modular group is called a modular function oran elliptic modular function.

Now consider what happens for some other unimodular matrices which are not con-gruent to the identity mod2. This will yield other functional equations for λ in addition tothe fact that λ is periodic of period 2. As before, these functional equations come aboutbecause ℘ is unchanged when you change the basis for M, the module of periods. Inparticular, consider the unimodular matrices

(1 01 1

),

(0 11 0

). (58.1.15)

Consider the first of these. Thus

(w′1w′2

)=

(w1

w1 +w2

)