  Argyres and co-workers started a program to classify all 4d $\mathcal{N}=2$
QFT by classifying Special Geometries with appropriate properties. They
completed the program in rank-1. Rank-1 $\mathcal{N}=2$ QFT are equivalently
classified by the Mordell-Weil groups of certain rational elliptic surfaces.
The classification of 4d $\mathcal{N}=2$ QFT is also conjectured to be
equivalent to the representation theoretic (RT) classification of all
2-Calabi-Yau categories with suitable properties. Since the RT approach smells
to be much simpler than the Special-Geometric one, it is worthwhile to check
this expectation by reproducing the rank-1 result from the RT side. This is the
main purpose of the present paper. Along the route we clarify several issues
and learn new details about the rank-1 SCFT. In particular, we relate the
rank-1 classification to mirror symmetry for Fano surfaces. In the follow-up
paper we apply the RT methods to higher rank 4d $\mathcal{N}=2$ SCFT.
