It is one of the first theorems proved in the theory of modular forms [1, p. 78] that every element in SL (2, ℤ), and hence in SL(2, ), can be written as a product of the two generators
While this is quite an easy result, it required more advanced tools such as spectral analysis on Riemann surfaces [2] to show that there is in fact always such a product of length O(log p). This result is not constructive and therefore immediately [2, p. 102] raises the following:
Problem. Is there an algorithm polynomial in log p for constructing a monomial in S and T, of degree O(log p), whose value is
In this note, we construct such an algorithm. We will carry out all the details only for the particular element
but it will become obvious that our method works for all elements in SL(2, ).