higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Arithmetic geometry is a branch of algebraic geometry studying schemes (usually of finite type) over the spectrum $Spec \mathbb{Z}$ of the ring of integers, for number theoretic purposes; usually one complements this with some data “at the prime at infinity” leading to a more modern notion of an arithmetic scheme (cf. Arakelov geometry).
wikipedia: glossary of arithmetic and Diophantine geometry, Arakelov geometry
“Arakelov geometry preprint arxiv”, list of links
C. Soulé, D. Abramovich, J. F. Burnol, J. K. Kramer, Lectures on Arakelov Geometry, Cambridge Studies in Advanced Mathematics 33, 188 pp.
conferences in arithmetic geometry, at Kiran Kedlaya’s wiki