We describe some periodic structure in the cohomology of the moduli stack of 1-dimensional formal group laws, also known as the $E_2$-page of the classical Adams--Novikov spectral sequence. This structure is distinct from the familiar $v_n$-periodicities, and it displays interesting number-theoretic properties. Our techniques involve the $\mathbb{ C}$-motivic Adams spectral sequence, and we obtain analogous periodic structure in $\mathbb{ C}$-motivic stable homotopy.
Categories: math.AT