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I am interested in pursuing an understanding of K-theory. Primarily, the $K_3(\mathbb{Z})$ algebraic K-group over ring of integers of an algebraic number field and its relationship to the $\mathbb{Z}/48$ ring of integers modulo 48.

This is (of course), again, from Terry Gannon's "Moonshine Beyond the Monster" where he talks about many amazing coincidences with the number 24, the Riemann Zeta Function $\Sigma_{n=1}^\infty (1/n)^{-1} = -1/12$, Apery's constant, where $\Sigma_{n=1}^\infty (1/n)^2 = \pi/6$ (which he states are both synonomous in their relationship to $K_3(\mathbb{Z})\leftrightarrow \mathbb{Z}/48$....)

A little harder to discern is the (possible) relationship of the Bimonster, $(M \times M) \rtimes \mathbb{Z}/2 \to M \wr 2$, and the Incidence Graph of the M-13 pseudogroup with 13 points and lines, (the 13 point, 13 line projective plane, where here, the coincidence would appear to be the number 26, which is the dimension of Bosonic String Theory (2 + 24 dimensions, the quantum harmonic oscillator on a 2-dimbrane, which relates to -1/12 above per John Baez "My Favorite Number is 24")). It's tempting to see the resemblance of 24 relating to the Monster, and 48 to the BiMonster, but that seems to obvious. Finally, is there any relevance in bringing in the M12-Mathieu group here, being so close to the M13-pseudogroup? I apologize ahead of time if this last paragraph is "shooting the moon" but hopefully my first two paragraphs are well-stated questions.

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I've taken the liberty of TeX-ing the question, but I couldn't figure out what the '.x' in 'M X M .x Z/2' is. – David Roberts Nov 9 '10 at 1:14
Actually I'll take a stab and guess that it is semi-direct product. – David Roberts Nov 9 '10 at 1:15
"...but hopefully my first two paragraphs are well-stated questions." Perhaps it's just me, but I wasn't able to find any questions in the first two paragraphs, well-stated or otherwise. – Pete L. Clark Nov 9 '10 at 3:23
It seems a bit strange to me that you bring up a "relationship" between $\mathbb{Z}/48$ and $K_3(\mathbb{Z})$, since it is both clearer and stronger to point out that the two groups are isomorphic. – S. Carnahan Nov 9 '10 at 14:28
up vote 8 down vote accepted

I am not sure if this helps you since it doesn't contain any speculations on connections with finite simple groups, but the torsion in $K_{2n+1}(\mathcal{O}_F)$ is now fairly well understood for arbitrary $n$ and for rings of integers $\mathcal{O}$ of arbitrary number fields, thanks to Suslin, Voevodsky, Rost, et al. The torsion subgroups are essentially certain Tate modules, not only as groups but in fact as Galois modules. In particular, the way 48 appears in the description of $K_3(\mathbb{Z})$ is that it happens to be twice the biggest $n$, such that the exponent of $(\mathbb{Z}/n\mathbb{Z})^\times$ divides 2, and this observation fits into a bigger picture.

For a sketch of this bigger picture, see e.g. the article by Charles Weibel "Algebraic K-Theory of Rings of Integers in Local and Global Fields" in the Handbook of K-theory. He also gives references for the proofs of the various known results.

Edit: I should also mention that the connection with the Riemann zeta function (or, in the case of number fields, with the Dedekind zeta function of the number field), is - at least conjecturally - not a coincidence either. The special values of Dedekind zeta functions are conjecturally described by sizes of torsion subgroups and volumes of free parts of higher $K$-groups of the rings of integers. This is the Lichtenbaum conjecture. That in turn is also part of a bigger conjectural picture, the Bloch-Kato conjecture on Tamagawa numbers. In short: the connection that you are asking about is a special case of a large conjectural framework of global-local principles, which all take their cue from the analytic class number formula, but are completely out of reach at present.

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