Gauss–Manin connection in disguise: Dwork family

Ramanujan systems of Rankin-Cohen type and hyperbolic triangles

“…Movasati [17] proved that the Ramanujan system (1.1) is the unique vector field on the moduli space of the family of the elliptic curves (1.2) y 2 = 4(x − t 1 ) 3 − t 2 (x − t 1 ) − t 3 , (t 1 , t 2 , t 3 ) ∈ C 3 with 27t 2 3 − t 3 2 = 0 , that satisfies a certain equation involving the Gauss-Manin connection of the universal family of (1.2). Pursuing this interpretation, Movasati [18], [19] introduced a new technique called Gauss-Manin connection in disguise and, together with the second author [16], used it to associate a canonical system of non-linear ODEs to a moduli space of enhanced Calabi-Yau n-folds arising from the Dwork family, for any n ∈ N. An interesting arithmetic aspect of their work is the following. The solutions of these systems for n = 1, 2 are quasimodular forms on congruence subgroups; for n = 3, 4, the q-expansion solutions have integral coefficients but, due to their fast growth, can not be related to classical quasimodular forms.…”