2014/09/23 by James Newton, Newton, James
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebra representation #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Automorphic form #Conjecture #Counterexample #Discrete mathematics #Division algebra #FOS: Mathematics #Galois module #Mathematics #Modular form #Number Theory (math.NT) #Pure mathematics #Quaternion algebra #Raising (metalworking) #math.NT
paper · pdf · open access · doi:10.48550/arxiv.1409.6533
published in Research Portal (King's College London) (King's College London) · 24 pages
arxiv created 2014/09/23 · openalex publication_date 2014/09/23 · arxiv updated 2014/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.