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arXiv · 1007.0181

Higher congruence companion forms

Rajender Adibhatla·Jayanta Manoharmayum
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Abstract

For a rational prime $p \geq 3$ we consider $p$-ordinary, Hilbert modular newforms $f$ of weight $k\geq 2$ with associated $p$-adic Galois representations $ρ_f$ and $\mod{p^n}$ reductions $ρ_{f,n}$. Under suitable hypotheses on the size of the image, we use deformation theory and modularity lifting to show that if the restrictions of $ρ_{f,n} $ to decomposition groups above $p$ split then $f$ has a companion form $g$ modulo $p^n$ (in the sense that $ρ_{f,n}\sim ρ_{g,n}\otimesχ^{k-1}$).

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Rajender Adibhatla, Jayanta Manoharmayum. 2013-04-11. Higher congruence companion forms. https://doi.org/10.4064/aa156-2-4

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