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It is well known [Clausen, p-adic J-homom., in introduction] that there are cyclic subgroups of $\pi_{4k-1}S \: (k>o)$ with size the zeta values $B_{2k}/k \: (=-\zeta(1-2k))$ which completely account for the first chromatic layer of $\pi_*S$ at odd primes.

Question: is it possible to describe the higher layers in the chromatic filtration in the stable homotopy groups of spheres in terms of Bernoulli numbers?

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    $\begingroup$ There is no reason to think that that would be true. The appearance of Bernoulli numbers is tightly connected to $K$-theory and the multiplicative group, which are only relevant at height one. There is no sign of Bernoulli numbers in the known calculations at height two. $\endgroup$
    – Neil Strickland
    Commented Jul 26, 2022 at 17:33
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    $\begingroup$ I don't know much about this so take it with a grain of salt, but if I remember correctly, the second layer can be described in terms of certain modular forms, specifically cusp forms, by work of Mark Behrens. This description also includes the first layer in terms of the Eisenstein series $E_{2k}$. Descending from this description to the Bernoulli number description corresponds to setting $q=0$ in the $q$-expansion of the modular form, since the constant term of the Eisenstein series $E_{2k}$ (suitably normalized) is precisely $-B_{2k}/4k$. $\endgroup$
    – pregunton
    Commented Jul 26, 2022 at 20:14
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    $\begingroup$ But doing that to the cusp forms one just gets $0$, so that would be an indication that there aren't any single numbers associated to the second layer in the same sense that Bernoulli numbers are associated to the first layer. $\endgroup$
    – pregunton
    Commented Jul 26, 2022 at 20:15

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