About some family of elliptic curves

Abstract

We examine the moduli space E=T* of complex tori T(t)=C/L(t) where L(t)=cost.n(t)Lt. We find that the Dedekind eta function furnishes a bridge between the euclidean and hyperbolic structures on T*=C-L/L as well as between the doubly periodic Weierstrass function p on T* and the theta function for the lattice E(8). The former one allows us to rewrite the Lame equation for the Bers embedding of T(1,1) in a new form. We show that L has natural decomposition into 8 sublattices (each equivalent to L) together with appropriate half-points and that this leads to some local functions and to a relation with E(8)