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 $\mathrm{\Delta }$ be a subgroup of $(A\mathrm{/}(\mathfrak{n}){)}^{\times }$ such that the map $\mathrm{\Delta }\to (A\mathrm{/}(\mathfrak{n}){)}^{\times }\mathrm{/}{\mathbb{F}}_q^{\times }$ is bijective. Let $S$ be a scheme over $A[1\mathrm{/}\mathfrak{n}]$ and let $R$ be an $A[1\mathrm{/}\mathfrak{n}]$-algebra which is an excellent regular domain. In this paper, we show that any Drinfeld module $E$ of rank two over $S$ admitting a ${\mathrm{\Gamma }}_1^{\mathrm{\Delta }}(\mathfrak{n})$-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 ${\mathrm{\Gamma }}_1^{\mathrm{\Delta }}(\mathfrak{n})$ over $R$, we give a dual Kodaira–Spencer isomorphism of the form ${\bar{\omega }}^{\otimes 2}\simeq {\mathrm{\Omega }}_{X\mathrm{/}R}^1(2\mathrm{C}\mathrm{u}\mathrm{s}\mathrm{p}\mathrm{s})$, in contrast with the usual one in the Drinfeld case in which $E^D$ is involved.