New approach to Calabi-Yau operators

Pencils of Calabi-Yau threefolds give rise to local systems and differential operators with very strong arithmetical properties. One interesting source of examples come from regularised quantum cohomology D-modules of Fano varieties. Although no classification is known, one may  hope to characerise and construct such operators  from  first principles.
I will describe joint work in progress with V. Golyshev and A. Mellit, which shows that a Langlands  approch via an explicit description of a Hecke-algebra given by Kontsevich,  combined with the new idea of a "congruence sheaf" leads to a practical approach  in the rank two case. This leads to a new approach to obtain the Apery-Beukers-Zagier  operators of the Beauville list of elliptic modular surfaces.