Computation of the Teichmüller distance between elliptic curves

E. Klassen, C. Nolder, M. Seppälä, T. Sutton

Complex algebraic genus one curves can be uniformized by elliptic integrals. This is both classical and explicit. For any genus one curve $C$ defined by an equation $y^2 = x(x-1)(x-\lambda)$ one can explicitly form a lattice $L = \langle z\mapsto z+\omega_1, z\mapsto z+\omega_2\rangle$ such that $C = \C/L$. One can, furthermore, find the uniformizing projection $\C\to C$ (\cite{Siegel:I}). In this note that fact is used to find Teichm\"uller mappings between two given genus one algebraic curves. In fact, for any two elliptic curves given by their defining polynomials it is possible to find all the Teichm\"uller mappings between them. One can, furthermore, compute the Teichm\"uller distance between given elliptic curves (in the moduli space).