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.

A description is given of all primitive δ-series mod p of order 1 which are eigenvectors of all the Hecke operators nTκ(n), "pTκ(p)", (n, p) = 1, and which are δ-Fourier expansions of δ-modular forms of arbitrary order and weight w with deg(w) = κ ≥ 0; this set of δ-series is shown to be in a natural one-to-one correspondence with the set of series mod p (of order 0) which are eigenvectors of all the Hecke operators T κ+2 (n), T κ+2 (p), (n, p) = 1 and which are Fourier expansions of (classical) modular forms of weight ≡ κ+2 mod p−1.

We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation". This property can be viewed as a special kind of overconvergence property. One can also go in the opposite direction by using differential functions that arise in a ramified situation to construct "new" (unramified) differential functions.

Using Buium's theory of arithmetic differential characters, we construct a filtered F -isocrystal H(A) K associated to an abelian scheme A over a p-adically complete discrete valuation ring with perfect residue field. As a filtered vector space, H(A) K admits a natural map to the usual de Rham cohomology of A, but the Frobenius operator comes from arithmetic differential theory and is not the same as the usual crystalline one. When A is an elliptic curve, we show that H(A) K has a natural integral model H(A), which implies an integral refinement of a result of Buium's on arithmetic differential characters. The weak admissibility of H(A) K depends on the invertibility of an arithmetic-differential modular parameter. Thus the Fontaine functor associates to suitably generic A a local Galois representation of an apparently new kind.

We introduce differential characters of Drinfeld modules. These are function-field analogues of Buium's p-adic differential characters of elliptic curves and of Manin's differential characters of elliptic curves in differential algebra, both of which have had notable Diophantine applications. We determine the structure of the group of differential characters. This shows the existence of a family of interesting differential modular functions on the moduli of Drinfeld modules. It also leads to a canonical F -crystal equipped with a map to the de Rham cohomology of the Drinfeld module. This F -crystal is of a differential-algebraic nature, and the relation to the classical cohomological realizations is presently not clear.