We investigate the arithmetic of p-adic automorphic forms for certain quaternion algebras
over totally real fields (including GL2/Q), focussing on the question of the relationships
between p-adic automorphic forms with different levels at a prime l different from p, and
the relation between p-adic automorphic forms for (the multiplicative groups of) different
quaternion algebras (i.e. a p-adic Jacquet–Langlands correspondence). In chapters 2 and 3
we prove results of ‘level raising’ type, showing that certain families of p-adic automorphic
forms with level prime to l (l-old forms) intersect with a family of p-adic automorphic
form with Iwahori level at l, where all the classical points in this second family are l-new.
Chapter 2 works with definite quaternion algebras over Q, whilst chapter 3 works with
GL2/Q. In chapter 3 the main tool is Emerton’s theory of completed cohomology. In
chapter 4 we study indefinite quaternion algebras over totally real fields F, split at one
infinite place, and prove level raising and lowering results. Finally, also in chapter 4, we
give an example of a cohomological construction of p-adic Jacquet-Langlands functoriality,
using completed cohomology.