Modular D3 equations and spectral elliptic curves

Determinantal differential equations were introduced by Vasily Golyshev and Jan Stienstra around 2005. The motivation comes from mirror symmetry for Fano varieties. I will talk about our recent work with Vasily on such equations of orders 2 and 3, that is D2 and D3. We show that the expansion of the analytic solution of a non-degenerate modular equation of type D3 over the rational numbers with respect to the natural parameter coincides, under certain assumptions, with the $q$-expansion of thenewform of its spectral elliptic curve and therefore possesses a multiplicativity property. We compute the complete list of D3 equations with this multiplicativity property and relate it to Zagier's list of non-degenerate modular D2 equations.