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

Abstract

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 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.

Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic

“…They also showed that it is realized as the Galois action on 4-torsion points on the elliptic curve E : Y 2 = X 3 + X 2 + X + 1, which is a modular elliptic curve of conductor 128 associated with a cusp form g of weight 2 and level 128; the cusp forms f, g are congruent mod 4. (See [8,Theorem 1].) Since F 4 ∼ = X 0 (64), it is interesting to study the relation between the Galois action on Jac(F 4 ) [4] and the 2-dimensional mod 4 Galois representation constructed by Kiming-Rustom.…”

“…Remark 5.2. The dihedral extension Q(2 1/4 , ζ 8 )/Q appeared in a recent work of Kiming-Rustom [8]. They constructed a 2-dimensional mod 4 Galois representation which factors through Gal(Q(2 1/4 , ζ 8 )/Q), and showed that it is associated with a cusp form f of weight 36 and level 1.…”