Higher congruence companion forms

Abstract

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

13 pages

Cited by in corpus (1)

• On certain finiteness questions in the arithmetic of modular forms