Norm and trace of the \(j\)-invariants of Drinfeld modules associated to hyperelliptic curves (Q677616)

Let \({\mathbf A}=\mathbb{F}_q[T]\) and \({\mathbf k}=\mathbb{F}_q(T)\), where \(\mathbb{F}_q\) is the finite field with \(q\)-elements. Let \(L/{\mathbf k}\) be a quadratic extension where the prime \(\infty\) of \({\mathbf k}\) ramifies and let \({\mathcal O}\supset{\mathbf A}\) be the ring of \({\mathbf A}\)-integers. A Drinfeld \({\mathcal O}\)-module \(\psi\) of rank 1 is also a Drinfeld \({\mathbf A}\)-module of rank 2. As such it is analogous to an elliptic curve with complex multiplication. In this paper, the authors compute the degrees of the norm and trace of \(j(\psi)\) generalizing earlier results where \(L\) had genus 1.