The group SL(2,ℤ) of 2x2 matrices with integer entries and unit determinant acts naturally on the upper half-plane by fractional linear transformations. Identifying points in the upper half-plane which are in the same SL(2,ℤ)-orbit yields a space whose points can be interpreted as isomorphism classes of elliptic curves.
In this project you will study this space in some detail, paying particular attention to the modular functions: meromorphic functions defined on the upper half-plane which transform "nicely" under SL(2,ℤ). The project is based on Chapter VII of Serre's book (with the possible omission of Section 5 on Hecke operators).
Individual project for Single Degree students in Mathematics