Let me admit right at the outset that I have a very superficial outsider's knowledge of homotopy theory. Nevertheless, I was trying to gain some understanding of Hopkins' ICM lecture 'algebraic topology and modular forms.'

In section 6, he mentions two constructions. To a map

$$\phi: MSpin\rightarrow KO$$

of $E_{\infty}$ ring spectra, he associates a characteristic power series $$K_{\phi}(x)\in \mathbb{Q}[[x]].$$ Similarly, to an $E_{\infty}$-map

$$\psi: MO\langle 8\rangle \rightarrow tmf,$$

he associates a power series $$K_{\psi}(x)\in MF_{\mathbb{Q}}[[x]],$$ where $tmf$ is the topological modular form spectrum and $MF_{\mathbb{Q}}=MF\otimes _{\mathbb Z}\mathbb Q$ is the ring of modular forms with rational coefficients.

I wonder if someone could give a brief outline of how these associations are carried out. I presume it is something elementary having to do with the homotopy groups of $MSpin$ and $MO\langle 8\rangle$, but I don't quite have the resources right now to track these down.

As usual with questions of this sort, I'm sure my level of ignorance is incongruous with the words I am employing already, but thank you in advance for any tolerant answers or references.

Added:

Maybe I should summarize the point of this question for fellow number-theorists who are too busy to look into the paper. In the notation above, one associates to $\phi$ a characteristic sequence

$$b(\phi)=(b_2, b_4, b_6,\ldots)$$

via the formula

$$\log(K_{\phi}(x))=-2\sum_{n>0} b_n\frac{x^n}{n!}.$$

Incredibly, this procedure sets up a bijection:

homotopy classes of $E_{\infty}$ maps from $MSpin$ to $KO$ $\leftrightarrow$ the set of sequences of rational numbers $(b_i)$ as above that satisfy

(1) $b_n\equiv B_{n}/n \ \ \mod \mathbb{Z}$, where the $B_n$ are the Bernouilli numbers;

(2) for each odd prime $p$ and $p$-adic unit $c$,

$$m\equiv n \ \mod p^k(p-1) \Rightarrow (1-c^n)(1-p^{n-1})b_n \equiv (1-c^m)(1-p^{m-1})b_m \ \mod p^{k+1};$$

(3) for each $2$-adic unit $c$,

$$m\equiv n \ \mod 2^k \Rightarrow (1-c^n)(1-2^{n-1})b_n \equiv (1-c^m)(1-2^{m-1})b_m \ \mod 2^{k+2}.$$

In the case of the homotopy classes of maps from $MO\langle 8\rangle$ to $tmf$, one gets similar congruences involving Eisenstein series instead of their constant terms. Incidentally, perhaps these congruences imply the ones above?