Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 057, 13 pages      arXiv:1802.04976
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

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

Ian Kiming a and Nadim Rustom b
a) Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
b) Department of Mathematics, Koç University, Rumelifeneri Yolu, 34450, Sariyer, Istanbul, Turkey

Received February 28, 2018, in final form June 04, 2018; Published online June 13, 2018

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.

Key words: congruences between modular forms; Galois representations.

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