Riemann Surfaces and Modular Function Field Extensions

Bryden Cais

This is my undergraduate senior honors thesis written under the supervision of Noam Elkies.

For precisely the values N=1,2,3,4,5 the modular curve X(N) has genus 0, and the field of (meromorphic) modular functions of level N is rational, generated (over C) by a single function j_N, which is unique up to projective automorphisms. We use Klein forms to determine j_N explicitly (as a q-product). We then study the Galois covering of curves X(N)-->X(1) with Galois group G_N=PSL_2(Z/NZ). In particular, for a given index N subgroup of G_N, there is a subfield K_N of K(X(N)) of degree N over K(X(1)). By the primitive element theorem, K_N is generated over C(j_1) by a single element, and we use algebraic geometry to write down a polynomial over C(j_1) having such an element as a root.