Arithmetic gravity: An approach to adelic physics via algebraic spaces

In this paper we present an approach to adelic physics via algebraic spaces. Relative algebraic spaces X → S are considered as fundamental objects which describe space-time. This yields a number field invariant formulation of general relativity which, in the special case S = Spec ℂ, may be translated back into the language of manifolds. With regard to adelic physics the case of an excellent Dedekind scheme S as base scheme is of interest (e.g. S = Spec ℤ). Some solutions of the arithmetic Einstein equations are studied.