Drinfeld module and Weil pairing over Dedekind domain of class number two

The primary objective of this paper is to derive explicit formulas for rank one and rank two Drinfeld modules over a specific domain denoted by $A$. This domain corresponds to the projective line associated with an infinite place of degree two. To achieve the goals, we construct a pair of standard rank one Drinfeld modules whose coefficients are in the Hilbert class field of $A$. We demonstrate that the period lattice of the exponential functions corresponding to both modules behaves similarly to the period lattice of the Carlitz module, the standard rank one Drinfeld module defined over rational function fields. Moreover, we employ Anderson's t-motive to obtain the complete family of rank two Drinfeld modules. This family is parameterized by the invariant $J={\lambda }^{q^2+1}$ which effectively serves as the counterpart of the j-invariant for elliptic curves. Building upon the concepts introduced by van der Heiden, particularly with regard to rank two Drinfeld modules, we are able to reformulate the Weil pairing of Drinfeld modules of any rank using a specialized polynomial in multiple variables known as the Weil operator. As an illustrative example, we provide a detailed examination of a more explicit formula for the Weil pairing and the Weil operator of rank two Drinfeld modules over the domain $A$.