On nearly holomorphic Drinfeld modular forms for admissible coefficient rings

Let X be a smooth projective and geometrically irreducible curve over the finite field Fq with q elements and K be its function field. Let ∞ be a fixed closed point on X and A be the ring of functions regular away from ∞. In the present paper, by generalizing the previous work of Chen and the first author, we introduce the notion of nearly holomorphic Drinfeld modular forms for congruence subgroups of GL₂(K) as continuous but non-holomorphic functions on a certain subdomain of the Drinfeld upper half plane. By extending the de Rham sheaf to a compactification MI²̄ of the Drinfeld moduli space MI², we also describe such forms algebraically as global sections of an explicitly described sheaf on MI²̄ as well as construct a comparison isomorphism between analytic and algebraic description of them. Furthermore, we show the transcendence of special values of nearly holomorphic Drinfeld modular forms at CM points and relate them to the periods of CM Drinfeld A-modules.