See my own email to Magnus maybe.
Arxiv: Title: An archimedian analog of Iwasawa theory Authors: Ken-ichi Sugiyama
Read everything by Borger. For example: Big de Rham-Witt cohomology
Elliptic Bloch groups (what are these, and should they be here or moved to the motivic cohomology heading?)
Elliptic cohomology and all of classical chromatic homotopy theory!
Topological automorphic forms, Topological modular forms
Topological Langlands???
Must mention the remark on arithmetic index theory, and Grojnowski’s jumping point.
nLab page on N50 Arithmetic mysteries