2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77908Motivated by a conjecture of Lian and Yau concerning the mirror map in string theory, we determine when the mirror map q-series of certain elliptic curve and K3 surface families are Hauptmoduln (genus zero modular functions). Our geometric criterion for modularity characterizes orbifold uniformization properties of their Picard-Fuchs equations, effectively demystifying the mirror-moonshine phenomenon. A longer, more comprehensive treatment of these results will appear shortly.Submitted to the Proceedings of NATO-ASI and CRM Summer School on the Arithmetic and Geometry of Algebraic CyclesAlgebraic GeometryHigh Energy Physics - TheoryNumber Theory14D05, 11F03, 14J32 (Primary); 30F35, 14J28, 83E30 (Secondary)text