Special arithmetic of flavor

We revisit the classification of rank-1 4d 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-heron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs (epsilon, F-infinity) where epsilon is a relatively minimal, rational elliptic surface with section, and F(infinity )a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on (epsilon, F-infinity) equivalent to the safely irrelevant conjecture. The Mordell-Weil group of epsilon (with the Neron-Tate pairing) contains a canonical root system arising from (-1)-curves in special position in the Neron-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.