# Why do these two Monster-related calculations yield $163$?

Fact 1: (1979, Conway and Norton)$^{1}$

"There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster."

Note: There are 194 (linear) irreducible representations of $\mathbb{M}$, hence 194 conjugacy classes. Of these, there are 22 that are complex quadratic valued. Of the remaining, there are 9 linear dependencies.$^{2}$

Fact 2: (2004, C. Cummins)$^{3}$

"The genus 0 moonshine groups have $132 + 1 + 4 + 5 + 13 + 1 + 7=\color{blue}{163}$ equivalence classes with period 1."

Note: There are exactly $6486$ genus $0$ moonshine groups, but only $371$ equivalence classes. Of these, $310\,$ have a rational representative. (Curiously, $\color{red}{310} =163+67+43+19+11+7$.)

\begin{array}{|c|r|c|c|c|c|c|c|c|c|c|c|} \hline \text{Period}& & 15^* & 11^*& 7^*& 3^*& 2^*& i& i\,2& 2 & 5 &\color{brown}{\text{Total}}\\ \hline 1&132& 0& 1& 4& 5& 0& 13& 1& 0& 7 & \color{blue}{163}\\ 2&120& 0& 0& 2& 1& 7& 4& 2& 2& 2& 140\\ 3& 26& 1& 0& 1& 4& 0& 2& 0& 0& 0& 34\\ 4& 16& 0& 0& 0& 0& 2& 0& 0& 0& 0& 18\\ 6& 10& 0& 0& 0& 0& 0& 0& 0& 0& 0& 10\\ 8& 3& 0& 0& 0& 0& 0& 0& 0& 0& 0& 3\\ 12& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2\\ 24& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ \hline \color{brown}{\text{Total}}&\color{red}{310}& 1& 1& 7& 10& 9& 19& 3& 2& 9& \color{brown}{371}\\ \hline \end{array}

Question: While the value $163$ was calculated differently in Facts 1 and 2, are they just different ways of saying the same thing, and that one implies the other?

References:

1. Monstrous Moonshine, by J.H. Conway & S.P. Norton.
2. Sporadic and Exceptional, by Yang-Hui He & John McKay.
3. Congruence Subgroups of Groups Commensurable with PSL (2, Z) of Genus 0 and 1, by C. J. Cummins.
4. Tables by C. J. Cummins.
• The above is Table 5: Summary of genus 0 moonshine groups data by C.J. Cummins. – Tito Piezas III Mar 29 '18 at 14:53
• It would nevertheless be good to have the monster tag here. – F. C. Mar 30 '18 at 6:43
• @F.C. It is done. – Tito Piezas III Mar 30 '18 at 7:45
• Are there similar counts for the corresponding groups related to 43 and 67? – Wolfgang Apr 3 '18 at 20:32
• @Wolfgang: Of the 26 sporadic groups, the Monster contains 19 others as subquotients. Of the remaining 6, curiously the largest prime $p$ that divides the order of the Lyons group is $67$, for the Janko group $J_3$ is $43$, and $J_2$ is $19$. In Conway and Norton's paper, they asked "if there a similar period three automorphism for the case $\Gamma(67)+$". – Tito Piezas III Apr 4 '18 at 10:25

## 1 Answer

If there is a connection, I would expect it to be related to the fact that the $j$ invariant is connected to both Moonshine and to class field theory. Note that 7, 11, 19, 43, 67, and 163 are all the absolute values of odd discriminants with class number 1. (Strangely, 3 is missing from the list! Otherwise the list is complete.)

• Perhaps we can surmise that 3 is exceptional since the j-invariant vanishes at $j\big(\tfrac{1+\sqrt{-3}}{2}\big)=0$. I've also come across some results in class field theory where it is required that $3 \nmid d$. – Tito Piezas III Mar 30 '18 at 0:47