Hidden Symmetries and j(\tau)

TL;DR

The paper reveals hidden symmetries in the modular j-invariant related to supersingular primes p, showing how a specific meromorphic function on X_0(p) transforms under Hecke operators, leading to new relations among coefficients.

Contribution

It introduces a unique meromorphic function on X_0(p) whose Hecke image relates to j( au), uncovering hidden symmetries and coefficient relations for supersingular primes.

Findings

Relation between G_p and j( au) via Hecke operators

Discovery of hidden symmetry groups with order dividing p

New coefficient relations for j(t) in supersingular cases

Abstract

We show that for supersingular prime p the image of a unique meromorphic function G_p on X_0(p) (of degree two, with polar divisor {[0]_0,[\infty]_0}) under a certain Hecke operator is equal to j(\tau) (up to some additional constant). This generates quantities of relations between the coefficients of j(t) and leads to some group of hidden symmetries whose order must be divided by p.