On autoduality of Drinfeld modules and Drinfeld modular forms

Let $\mathbb{F}_q$ be the field of $q$ elements and let $A=\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. Let $\mathfrak{n}\in A\setminus \mathbb{F}_q$ be a monic polynomial with a prime factor of degree prime to $q-1$. Let $\Delta$ be a subgroup of $(A/(\mathfrak{n}))^\times$ such that the map $\Delta\to (A/(\mathfrak{n}))^\times/\mathbb{F}_q^\times$ is bijective. Let $S$ be a scheme over $A[1/\mathfrak{n}]$ and let $R$ be an $A[1/\mathfrak{n}]$-algebra which is an excellent regular domain. In this paper, we show that any Drinfeld module $E$ of rank two over $S$ admitting a $\Gamma_1^\Delta(\mathfrak{n})$-structure is isomorphic to its Taguchi dual $E^D$. As an application, for the Hodge bundle $\bar{\omega}$ on the Drinfeld modular curve $X$ of level $\Gamma_1^\Delta(\mathfrak{n})$ over $R$, we give a dual Kodaira--Spencer isomorphism of the form $\bar{\omega}^{\otimes 2}\simeq \Omega^1_{X/R}(2\mathrm{Cusps})$, in contrast with the usual one in the Drinfeld case in which $E^D$ is involved.