In the 1960s, Novikov noticed that descent along the unit map for MU, the ring spectrum representing complex cobordism, was a particularly fruitful method for computing the homotopy groups of spheres. Within a few years, Quillen observed a connection between MU and geometric objects called formal groups. This was the beginning of chromatic homotopy theory and, among many other discoveries, revealed new Bott-periodicity-like symmetries in the homotopy groups of spectra. Since the 2000s, many of these results are best conceptually understood through the language of stacks in which they take on a rich arithmetic flavor. In this talk, we'll catch a glimpse of how the moduli stack of formal groups controls stable homotopy theory, and in particular how understanding its geometry offers conceptual explanations for these periodicity phenomena in the homotopy groups of spectra.