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arXiv · 1802.04976

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

Ian Kiming·Nadim Rustom
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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 ${\mathbb Q}$. 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.

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Ian Kiming, Nadim Rustom. 2018-06-13. Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4. https://doi.org/10.3842/sigma.2018.057

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