Effective conditional bounds on singular S-units.

We provide a proof that, for every prime $\ell \geq 5$, if $S_0$ is the set of all primes congruent to $1$ modulo $3$ and $S_\ell=S_0 \cup \{\ell \}$, then under certain assumptions on the $L$-functions attached to imaginary quadratic fields, the set of singular $S_\ell$-units is finite and its cardinality can be effectively bounded.