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

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 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 to illustrate certain questions that have arisen lately in the theory of modular forms modulo prime powers. We give a brief survey of those questions.