Arithmetic Levi-Civita connection

This paper is part of a series of papers where an arithmetic analogue of classical differential geometry is being developed. In this arithmetic differential geometry functions are replaced by integer numbers, derivations are replaced by Fermat quotient operators, and connections (respectively curvature) are replaced by certain adelic (respectively global) objects attached to symmetric matrices with integral coefficients. Previous papers were devoted to an arithmetic analogue of the Chern connection. The present paper is devoted to an arithmetic analogue of the Levi-Civita connection

The ring of differential Fourier expansions

For a fixed prime we prove structure theorems for the kernel and the image of the map that attaches to any differential modular function its differential Fourier expansion. The image of this map, which is the ring of differential Fourier expansions, plays the role of ring of functions on a "differential Igusa curve". Our constructions are then used to perform an analytic continuation between isogeny covariant differential modular forms on the differential Igusa curves belonging to different primes