The values of solutions of linear differential equations occasionally happen to be expressible via interesting arithmetic quantities, like the values of $L$-functions at integers. Several sources of this phenomenon are known, however most of the underlying identities remain unproven. In my talk I will systematically walk through examples of such identities linked with arithmetic differential equations of second order. Surprisingly enough, not all such second order instances are pullbacks of hypergeometric or Heun equations; these new arithmetic differential equations source from innocent-looking identities for $\pi$ and are a subject of study in our recent work with Mark van Hoeij and Duco van Straten.
Vladimir Rubtsov, Ilia Gaiur
