About some family of elliptic curves

TL;DR

This paper explores the moduli space of complex tori, revealing connections between elliptic functions, the Dedekind eta function, and lattice decompositions related to E8, offering new insights into their geometric and algebraic structures.

Contribution

It introduces a novel analysis of the moduli space of elliptic curves, linking classical functions with lattice decompositions and E8 structures.

Findings

Dedekind eta function bridges Euclidean and hyperbolic structures.

Reformulation of the Lame equation via the eta function.

Decomposition of lattice L into 8 sublattices related to E8.

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)