Chebyshev Covers and Exceptional Number Fields

Among the simplest of the classical polynomials are the Chebyshev polynomials of the first and second kind, Tk(x) and Uk(x). In our normalization, the indices are allowed to be halfintegers as well as integers, and the “polynomials” actually live in Z[x, √ 2− x, √ 2 + x]. We will show that the rational functions Tm/2(x) Tn/2(x) and Um/2(x) Un/2(x) are very remarkable from the point of view of Grothendieck’s dessins d’enfants. The fibers of these rational functions are likewise very remarkable from the point of view of algebraic number theory. For example, for (m,n) = (125, 128) the fiber of the second function above 5 is given by a degree 15875 polynomial in Z[x] with discriminant −2130729563437 and Galois group the entire symmetric group S15875.