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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2018
• Main Authors: Kiming, I., Rustom, N.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2018
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/209515
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4 / I. Kiming, N. Rustom // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 28 назв. — англ.

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.