**Title:** *p-adic interpolation of modular forms***Abstract:** The absolute Galois group G_**Q** of the rational numbers encodes the structure of all extensions of **Q** generated by solutions of polynomial equations. One way to explore this complicated group is via its representations, a large class of which can be produced from some complex analytic objects called modular forms. Identifying modular forms with points on a *p*-adic variety allows one to use geometric tools to gather insight into the Galois representations that they carry. Nowadays we know how to interpolate in this way all of the forms that are of finite slope, meaning that they are not in the kernel of a certain Hecke operator. I will explain how one could prove that interpolation is usually impossible if the slope is infinite.