One of the consequences of Wiles' proof of Fermat's Last Theorem is that elliptic curves over rational numbers can be associated with modular forms, whose Fourier coefficients essentially count points on the curve. Generalisation of this modularity to higher dimensional varieties is a very interesting open question. In this talk I will give a physicist's view of Calabi-Yau modularity. Starting with a very simplified overview of some number theoretic background related to the Langlands program, I relate some of this theory to black holes in IIB/A string theories compactified on Calabi-Yau threefolds. It is possible to associate modular forms to certain such black holes. We can then ask whether these modular forms have a physical interpretation as, for example, counting black hole microstates. In an attempt to answer this question, we derive a formula for fully instanton-corrected black hole entropy, which gives an interesting hint of this counting. The talk is partially based on recent work arXiv:2104.02718 with P. Candelas and J. McGovern.
