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.'](http://arxiv.org/abs/math/0212397) 


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?



Characteristic power series for maps of E_{\infty} ring spectra
In short, the series $K\_\phi$ is the "Hirzebruch characteristic series" which arises in the construction/calculation of genera, and in Hirzebruch-Riemann-Roch. The first few chapters of *Manifolds and modular forms* by Hirzebruch et al. describe the classical version of this pretty well.


If I have a one dimensional formal group law $F$ over a ring $A$, then over $A\_{\mathbb{Q}}$ there is an isomorphism $\mathrm{exp}\_F: G\_a\to F$ with the additive formal group. Let $K(x)=x/\mathrm{exp}\_F(x)$.


Now suppose $R$ is a "complex orientable cohomology theory", which means we are given a suitable isomorphism of rings $R^\*(CP^\infty)\approx \pi\_\*R[[x]]$. Such a theory has an associated formal group law $F$ (induced by the map $CP^\infty\times CP^\infty\to CP^\infty$ which classifies tensor product of line bundles), and thus there is an assocated series $K(x)=x/\mathrm{exp}\_F(x)$ in $\pi\_\*R\_\mathbb{Q}[[x]]$.


It turns out that a map of ring spectra $\phi:MU\to R$ corresponds exactly to giving a complex orientation of $R$. By Thom, elements of $\pi\_\*MU$ correspond to cobordism classes of stably-almost-complex manifolds, and there is a standard calculus due to Hirzebruch of calculating the effect of the map $\pi\_\*MU \to \pi\_\*R\_{\mathbb{Q}}$ using $K(x)$, which I might as well call $K\_\phi(x)$, since it depends on $\phi$. The formula (if I remember correctly), is that, if $[M]\in \pi\_\*MU$ is the class corresponding to a manifold of dimension $2n$, then 
$$
\phi(M) = \langle K\_\phi(x\_1)\dots K\_\phi(x\_n), [M] \rangle,
$$
where the $x\_i$ are the "chern roots" of the tangent bundle of $M$, and $[M]\in H\_{2n}M$ is the fundamental class.


There is a "universal example" of a $K\_\phi$, corresponding to the identity map $\phi\colon MU\to MU$. It turns out that $\pi\_\*MU\_{\mathbb{Q}}$ is a polynomial ring on the coefficients of $K\_\phi$, so that $K\_\phi(x)=\sum a\_{i-1}x^i$ (with $a\_0=1$) and $\pi\_\*MU\_{\mathbb{Q}}=\mathbb{Q}[a\_1,a\_2,\dots]$. (I'll need this later.)


In his talk, Mike isn't talking about complex orientations, but rather orientations with respect to $MSpin$ or $MO\langle 8\rangle$ (instead of $MO\langle 8\rangle$, we call it $MString$ these days, for some reason).


There is a map of ring spectrum $MU\to MSO$, induced by the apparent homomorphisms $U(n)\to SO(2n)$ of Lie groups. There is also a map $MSpin\to MSO$, induced by the double cover of lie groups. Although $MSpin\neq MSO$, we have that $\pi\_\*MSpin\_{\mathbb{Q}}\to \pi\_\*MSO\_{\mathbb{Q}}$ is an isomorphsism. Thus, a map $\phi\colon MSpin\to R$ induces
$$ \pi\_\*MU\_{\mathbb{Q}}\to \pi\_\*MSO\_{\mathbb{Q}}\approx \pi\_\*MSpin\_{\mathbb{Q}}\to \pi\_\*R\_{\mathbb{Q}},$$
and we can get $K\_\phi(x)$ from this.


The $MO\langle 8\rangle$ case is a little trickier.
There is a map $MU\langle 6\rangle \to MO\langle 8\rangle$, so a ring spectrum map $\phi\colon MO\langle 8\rangle\to R$ gives rise to a map
$$ MU\langle 6\rangle\_{\mathbb{Q}} \to R\_{\mathbb{Q}}.$$
On the other hand, the effect of the map $MU\langle 6\rangle\to MU$, on homotopy groups tensored with $\mathbb{Q}$, is
$$ \mathbb{Q}[a\_3,a\_4,\dots]\to \mathbb{Q}[a\_1,a\_2,a\_3,\dots].$$
So a map $\phi\colon MO\langle 8\rangle\to R$ gives us elements $\phi(a\_i)\in \pi\_{2i}R$ for $i\geq3$, which we can use as the coefficients of a series $K\_\phi(x)\in \pi\_\*R\_{\mathbb{Q}}$.


I must point out: there is actually an error in the statement of (2) and (3) given in Mike's talk. What he writes down are the "Kummer congruences"; but what one really needs to require are the "generalized Kummer congruences", which are basically the collection of all possible $p$-adic congruences involving Bernoulli numbers, not just the ones listed in (2) and (3). This comes from the theory of the "Mazur measure": the generalized Kummer congruences imply that the sequence $b\_n(1-p^{n-1})(1-c^n)$ can be interpolated to a function $f$, so that $f(n)$ for $n$ an integer is the moment of a measure on $\mathbb{Z}\_p^\times$. With (2) and (3) replaced by "interpolates to the moments of a measure on $\mathbb{Z}\_p^\times$", the result is correct.


Finally: There is a writeup of this at <http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf>, which may or may not be of any use to you!


