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We verify the (first) Gamma Conjecture, which relates the gamma class of a Fano variety to the asymptotics at infinity of the Frobenius solutions of its associated quantum differential equation, for all of the 17 deformation classes of rank one Fano 3-folds. Doing this involves computing the corresponding limits (“Frobenius limits”) for the Picard–Fuchs differential equations of Apéry type associated by mirror symmetry to the Fano families, and is achieved by two methods, one combinatorial and one using the modular properties of the differential equations. The Gamma Conjecture for Fano 3-folds always contains a rational multiple of the number ζ(3). We present numerical evidence suggesting that higher Frobenius limits of Apéry-like differential equations may be related to multiple zeta values.

Доказательство гамма-гипотезы для трехмерных многообразий Фано с решеткой Пикара ранга 1