Differential modular forms on Shimura curves, I

The quotient of a Shimura curve by the isogeny equivalence relation is not an object of algebraic geometry. The paper shows how this quotient space becomes a geometric object in a more general geometry obtained from 'usual algebraic geometry', by adjoining a new operation; this operation looks like a 'Fermat quotient' and should be viewed as an arithmetic analogue of usual derivations.