Parallelisms and Translations of (Affine) SL(2,q)-Unitals

Abstract

Unitals can be obtained as closures of affine unitals via parallelisms. The isomorphism type of the closure depends on the chosen parallelism, which need not be unique. For affine $\operatorname{SL}(2,q)$-unitals, we introduce a class of parallelisms for odd order and one for square order. Translations are automorphisms of unitals, fixing each block through a given center. For each of the known parallelisms of affine $\operatorname{SL}(2,q)$-unitals, we compute all possible translations with centers on the block at infinity.