Automorphic representations and their local factors
The fine geometric expansion of the Arthur-Selberg trace formula, which is a prerequisite for the global Jacquet-Langlands correspondence, shall be reformulated in a way that is valid in positive characteristic too.
We will study smooth representations of $GL_n(F)$, with $F$ a $p$-adic field, on $F_p$-vector spaces. We aim to generalise a representation theoretic construction which is available for $n=2$ to arbitrary $n$.
Periods of cuspidal automorphic representations of $GL_2$ and its inner forms at places of "split multiplicative type" shall be defined and their functorial properties and relations to $p$-adic $L$-functions and periods of $p$-adic Galois representations shall be studied.