K3 surfaces and modular forms
This talk concerns modular forms and their geometric realisations.
A classical construction of Shimura associates every Hecke eigenform
of weight 2 with rational coefficients to an elliptic curve over Q.
The converse statement that every elliptic curve over Q is modular, is
the Taniyama-Shimura-Weil conjecture, proven by Wiles, Taylor et al.
For higher weight, however, the opposite situation applies: Nowadays
we know the modularity for wide classes of varieties, but it is an
open problem whether all newforms of fixed weight with rational
coefficients can be realised in a single class of varieties.
I will present joint work with N. Elkies that provides the first
solution to the realisation problem in higher weight: We show that
every known newform of weight 3 with rational coefficients is
associated to a singular K3 surface over Q.