Drinfeld modules in rank 2 with CM and S-unit j-invariants

Abstract

We prove the finiteness of the set of $j$-invariants of Drinfeld modules of rank 2 over $\mathbb{F}_q[T]$ which are CM and $S$-units, for $S$ the infinite set of primes with even degrees. The proof is based on the study of ordinary reduction and supersingular reduction of Drinfeld modules, and on the splitting behaviour of primes dividing the difference of two Drinfeld singular moduli. We also provide an algorithm to compute a polynomial with coefficients in $\mathbb{F}_q[T]$ and roots the $j$-invariants having CM by a given order, and use it to compute some explicit examples, providing for instance counterexamples to a conjecture of Dorman. For a maximal order $\mathcal{O}$, we prove by a universality argument that our algorithm computes the Hilbert modular polynomial $H_\mathcal{O}$.