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

[摘要] 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.

[关键词] congruences between modular forms;Galois representations