The real J-homomorphism produces cyclic subgroups of size the denominator of $\zeta(1-2k)=-B_{2k}/k$ for $k>0$ in $\pi_{4k-1}S$ which completely account for first layer of the chromatic filtration on $\pi_*S$.

Question: Does the numerator in $\zeta(1-2k)$ also relate to $\pi_*S$ (at higher layers in the chromatic filtration)?

Guess: Can we think of denominator of $\zeta(1-2k)$ as local information (since it is hit by the J-homomorphism) and the numerator of $\zeta(1-2k)$ as global information?