| Summary: | Abstract We revisit the classification of rank-1 4d N=2 $$ \mathcal{N}=2 $$ QFTs in the spirit of Diophantine Geometry, viewing their special geometries as elliptic curves over the chiral ring (a Dedekind domain). The Kodaira-Néron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs (ε, F ∞) where E is a relatively minimal, rational elliptic surface with section, and F ∞ a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on (ε, F ∞) equivalent to the “safely irrelevant conjecture”. The Mordell-Weil group of E (with the Néron-Tate pairing) contains a canonical root system arising from (−1)-curves in special position in the Néron-Severi group. This canonical system is identified with the roots of the flavor group F: the allowed flavor groups are then read from the Oguiso-Shioda table of Mordell-Weil groups. Discrete gaugings correspond to base changes. Our results are consistent with previous work by Argyres et al.
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