Duality of Drinfeld modules and $\wp$-adic properties of Drinfeld modular forms

Abstract

Let $p$ be a rational prime and $q$ a power of $p$. Let $\wp$ be a monic irreducible polynomial of degree $d$ in ${\mathbf{F}}_q[t]$. In this paper, we define an analogue of the Hodge-Tate map which is suitable for the study of Drinfeld modules over ${\mathbf{F}}_q[t]$ and, using it, develop a geometric theory of $\wp$-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 $\infty$ modulo ${\wp }^n$, their weights are also congruent modulo $(q^d-1)p^{\lceil {\log }_p(n)\rceil }$, and that Drinfeld modular forms of level ${\Gamma }_1(\mathfrak{n})\cap {\Gamma }_0(\wp )$, weight $k$ and type $m$ are $\wp$-adic Drinfeld modular forms for any tame level $\mathfrak{n}$ with a prime factor of degree prime to $q-1$.

61 pages; Lemmata 3.3, 4.11 and 5.4 corrected, Section 4.5 added and corrected, other minor revisions

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