Abstract : The intimate relation between the arithmetic properties of varieties varying in families and the properties of the associated Picard-Fuchs differential is subject with a long and rich history that can be traced back to Deuring, Igusa, Dwork, Honda, Katz from which the notion of crystals emerged. A particular nice situation arises from families of Calabi-Yau motives, which can arise via various constructions, most notably via Mirror-Symmetry. In the two talks I will try to give a rough overview of this field, and illustrate it with specific examples. In particular, I will indicate how Calabi-Yau operators can be used to realise certain rank 4 motives attached Siegel paramodular forms by specific Calabi-Yau threefolds.

Keywords : arihmetic of differential equations