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Dihedral group, 4-torsion on an elliptic curve, and a peculiar eigenform modulo 4. (English) Zbl 1422.11103
Summary: 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.

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11F80 Galois representations
11G05 Elliptic curves over global fields
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LMFDB
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