Duality of Drinfeld modules and ℘ ‐adic properties of Drinfeld modular forms

Let p be a rational prime and q a power of p . Let ℘ be a monic irreducible polynomial of degree d inF q [ t ] . In this paper, we define an analogue of the Hodge–Tate map which is suitable for the study of Drinfeld modules overF q [ t ]and, using it, develop a geometric theory of ℘ ‐adic Drinfeld modular forms similar to Katz's theory in the case of elliptic modular forms. In particular, we show that for Drinfeld modular forms with congruent Fourier coefficients at ∞ modulo ℘ n , their weights are also congruent modulo( q d − 1 ) p ⌈ log p ( n ) ⌉, and that Drinfeld modular forms of levelΓ 1 ( n ) ∩ Γ 0 ( ℘ ) , weight k and type m are ℘ ‐adic Drinfeld modular forms for any tame level n with a prime factor of degree prime to q − 1 .