Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4

TL;DR

This paper presents a unique example of lifting a weak eigenform related to Ramanujan's tau function to a true eigenform over the rationals, highlighting new phenomena in modular forms modulo prime powers.

Contribution

It provides a concrete example of lifting a weak eigenform to a characteristic 0 eigenform, illustrating complexities in modular forms modulo prime powers.

Findings

The weak eigenform's coefficients match traces of Frobenius on 4-torsion of an elliptic curve.

The initial form cannot be lifted to a level 1 eigenform, showing limitations of classical liftability.

The example raises questions about modular forms modulo prime powers.

Abstract

We work out a non-trivial example of lifting a so-called weak eigenform to a true, characteristic 0 eigenform. The weak eigenform is closely related to Ramanujan's tau function whereas the characteristic 0 eigenform is attached to an elliptic curve defined over ${\mathbb Q}$. We produce the lift by showing that the coefficients of the initial, weak eigenform (almost all) occur as traces of Frobenii in the Galois representation on the 4-torsion of the elliptic curve. The example is remarkable as the initial form is known not to be liftable to any characteristic 0 eigenform of level 1. We use this example as illustrating certain questions that have arisen lately in the theory of modular forms modulo prime powers. We give a brief survey of those questions.