2014/09/23 by Newton, James
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1409.6533
This paper generalises previous work of the author to the setting of overconvergent p-adic automorphic forms for a definite quaternion algebra over a totally real field. We prove results which are analogues of classical `level raising' results in the theory of mod p modular forms. Roughly speaking, we show that an overconvergent eigenform whose associated local Galois representation at some auxiliary prime ł is (a twist of) a direct sum of trivial and cyclotomic characters lies in a family of eigenforms whose local Galois representation at ł is generically (a twist of) a ramified extension of trivial by cyclotomic. We give some explicit examples of p-adic automorphic forms to which our results apply, and give a general family of examples whose existence would follow from counterexamples to the Leopoldt conjecture for totally real fields. These results also play a technical role in other work of the author on the problem of local--global compatibility at Steinberg places for Hilbert modular forms of partial weight one.