### 0022: Part 1, The 163 Dimensions

Part 1: The 163 Dimensions of the Moonshine Functions

By Tito Piezas III

Keywords:  monstrous moonshine, moonshine functions, Monster group, prime-generating polynomials, McKay-Thompson series, pi formulas, 163.

Abstract:  An expository work on 1) Conway, Norton, and Atkin’s intriguing discovery that the moonshine functions span a linear space of dimension 163; and 2) the functions’ connection to Ramanujan-type pi formulas.

I. Introduction

II. The 172 McKay-Thompson Series and Linear Dependencies

III. More Pi Formulas?

IV. A Continued Fraction Using eπ√163

I.  Introduction

Consider the following prime-generating polynomials of form P(n) = an2-an+c,

P(n) = n2-n+41

P(n) = 2n2-2n+19

P(n) = 3n2-3n+23

P(n) = 4n2-4n+3

distinctly prime for n = {1 to c-1}, with the first (studied by Euler in 1772 in the form n2+n+41) for a remarkable 40 consecutive n.  And the following functions and their series,

1)  j(τ)  =  1/q + 744 + 196884q + 21493760q2 + 864299970q3 + …  (A007240)

2)  r2A(τ) = 1/q + 104 + 4372q + 96256q2 + 1240002q3 + …              (A101558)

3)  r3A(τ) = 1/q + 42 + 783q + 8672q2 + 65367q3 + …                        (A030197)

4)  r4A(τ) = 1/q + 24 + 276q + 2048q2 + 11202q3 + …                        (A097340)

q = e2πiτ = exp(2πiτ)

The first is the j-function j(τ), modular form of weight zero, but without the constant term, all these are simply the McKay-Thompson series T1A , T2A, T3A, and T4A of the Monster group.  Solving the polynomials for n at the point P(n) = 0, we get the complex roots,

n = (1+√-163)/2

n = (2+√-148)/4 = (1+√-37)/2

n = (3+√-267)/6

n = (4+√-32)/8 = (1+√-2)/2

where, consistent with the requirements for modular forms, we have chosen the root that is on the upper half of the complex plane. The first has fundamental discriminant -d = 163, the largest (in absolute value) with class number h(d) = 1, while the next two -d = {148, 267} = {4∙37,  3∙89} have class number h(d) = 2.  Let nτ, and plugging those into the appropriate series above, yields the integers,

1)  j((1+√-163)/2) = -6403203

2)  r2A((1+√-37)/2) = -141122

3)  r3A((3+√-267)/6) = -3003

4)  r4A((1+√-2)/2) = -27

The first appears in the Chudnovsky brothers’ pi formula,

where one can see the 163, as well as in the famous,

eπ√1636403203 + 743.9999999999992…

The second integer appears in a pi formula found by Ramanujan in 1912 (before the Monster group was even discovered),

where 4(37) also appears, as well as in the almost-integer,

eπ/2√148 = eπ√37141122 + 103.99997…

This (and similar formulas) inspired the one found by Chudnovsky brothers.  The third appears in another kind of pi formula,

where, one can see |d| = 267, and in the almost-integer,

eπ/3√267 = eπ√(89/3)3003 + 41.99997…

And finally, -27 appears in a fourth kind,

Using his own approach, Ramanujan found examples for all four types.  (See Pi Formulas and the Monster Group for the complete list of 36 known formulas where the denominator C is an integer.)  We instead used four functions based on T1A, T2A, T3A, T4A, but there are in fact many more of them.  And here’s a the astonishing fact first noticed by Conway and Norton:  the functions [1], [2], [3], [4], and others, call these the moonshine functions, span a linear space of dimension 163!  Interesting “coincidence” that it had to be that number, isn’t it?

II. The 172 McKay-Thompson Series and Linear Dependencies

To recall, the Monster group has 194 conjugacy classes so its character table has 194 rows and 194 columns.  Each column represents one conjugacy class and linear combinations of the entries gives its McKay-Thompson series which yields, to quote John McKay or Mark Ronan, a “moonshine function” (hence the title of this article).  There are 22 that are complex quadratic valued, so is reduced to 172 series, with the two distinct conjugacy classes A & B of order 27 still included as an exceptional case.  (Author note: My thanks to Michael Somos for clarifying certain points.)

 1 T1A 44 T12J 87 T25A 130 T46A 2 T2A 45 T13A 88 T26A 131 T46C 3 T2B 46 T13B 89 T26B 132 T47A 4 T3A 47 T14A 90 T27A 133 T48A 5 T3B 48 T14B 91 T27B 134 T50A 6 T3C 49 T14C 92 T28A 135 T51A 7 T4A 50 T15A 93 T28B 136 T52A 8 T4B 51 T15B 94 T28C 137 T52B 9 T4C 52 T15C 95 T28D 138 T54A 10 T4D 53 T15D 96 T29A 139 T55A 11 T5A 54 T16A 97 T30A 140 T56A 12 T5B 55 T16B 98 T30B 141 T56B 13 T6A 56 T16C 99 T30C 142 T57A 14 T6B 57 T17A 100 T30D 143 T59A 15 T6C 58 T18A 101 T30E 144 T60A 16 T6D 59 T18B 102 T30F 145 T60B 17 T6E 60 T18C 103 T30G 146 T60C 18 T6F 61 T18D 104 T31A 147 T60D 19 T7A 62 T18E 105 T32A 148 T60E 20 T7B 63 T19A 106 T32B 149 T60F 21 T8A 64 T20A 107 T33A 150 T62A 22 T8B 65 T20B 108 T33B 151 T66A 23 T8C 66 T20C 109 T34A 152 T66B 24 T8D 67 T20D 110 T35A 153 T68A 25 T8E 68 T20E 111 T35B 154 T69A 26 T8F 69 T20F 112 T36A 155 T70A 27 T9A 70 T21A 113 T36B 156 T70B 28 T9B 71 T21B 114 T36C 157 T71A 29 T10A 72 T21C 115 T36D 158 T78A 30 T10B 73 T21D 116 T38A 159 T78B 31 T10C 74 T22A 117 T39A 160 T84A 32 T10D 75 T22B 118 T39B 161 T84B 33 T10E 76 T23A 119 T39C 162 T84C 34 T11A 77 T24A 120 T40A 163 T87A 35 T12A 78 T24B 121 T40B 164 T88A 36 T12B 79 T24C 122 T40C 165 T92A 37 T12C 80 T24D 123 T41A 166 T93A 38 T12D 81 T24E 124 T42A 167 T94A 39 T12E 82 T24F 125 T42B 168 T95A 40 T12F 83 T24G 126 T42C 169 T104A 41 T12G 84 T24H 127 T42D 170 T105A 42 T12H 85 T24I 128 T44A 171 T110A 43 T12I 86 T24J 129 T45A 172 T119A

Table 1. The 172 McKay-Thompson series of Monstrous Moonshine

The first dozen coefficients of each of the 172 series are given in Table 4, Values of Head Characters, (p. 334) of John Conway and Simon Norton’s 1979 paper, Monstrous Moonshine.  Conveniently, an index of the McKay-Thompson series is also in the Online Encyclopedia of Integer Sequences.  (The moonshine functions, in accordance with Atlas notation, have the letter corresponding to the conjugacy class in upper-case, while the non-monstrous ones have it in lower-case.)  Also, David Madore has calculated the first 3500 coefficients of all the 172 moonshine McKay-Thompson series, while Michael Somos has an extensive database about the relationships between the coefficients and more.

But there are linear dependencies between the 172 series,

 # Linear dependency 1 T27A = T27B 2 T6A + 2T6E = T6B + T6C + T6D 3 T10A + 2T10E = T10B + T10C + T10D 4 T12A + 2T12I = T12B + T12E + T12H 5 T18B + 2T18D = T18A + T18C + T18E 6 T30B + 2T30G = T30A + T30C + T30F 7 2T8E = T4C + T4D 8 2T16B = T8D + T8E 9 T12C + T12E - T12F  - T12G - 2T12I = 2(T24B - T24C - T24I)

Table 2. The 9 Linear Dependencies

Taking into account these nine linear dependencies, then the dimension can be reduced to 172 – 9 = 163.  To bring this insight to life, in his blog, Lieven le Bruyn quotes p. 227 of Mark Ronan’s book Symmetry and the Monster,

Conway recalls that, ‘As we went down into the 160s, I said ‘...let’s guess what number we will reach.’ They guessed it would be 163 – which has a very special property in number theory – and it was! There is no explanation for this. We don’t know whether it is merely a coincidence, or something more....”

The first six were found by Conway and Norton, while the last three were discovered by Oliver Atkin who also showed that there are no more.  These nine are found in Conway and Norton’s paper, but in rather technical form.  For example, dependency # 7 is given as,

T4- + T4|2 = 2T8-

so I had to translate it into English, so to speak, i.e. convert it into the notation used in the OEIS (as well as in Madore’s and Somos’ databases).  We then find from the OEIS,

T4C(q) = 1/q + 20q - 62q3 + 216q5 - 641q7  + …      (A007248)

T4D(q) = 1/q - 12q + 66q3 - 232q5 + 639q7  + …      (A007249)

T8E(q) = 1/q  +  4q  +  2q3  -  8q5  - q7   + …            (A029841)

and we can easily see from the first few coefficients that indeed,

T4C + T4D = 2T8E

though it is proven by Atkin that the equality holds for all terms.  Thus, just as one can consider the series T27B as redundant being equal to T27A, and since there are nine linear dependencies, then only 163 series as “moonshine functions” are really necessary to linearly express all possible values of the 172 series.  Or, as was mentioned, these functions span a linear space of dimension 163.

In an email, McKay calls this as a “delicious coincidence”, while Norton opines this may be merely a coincidence.  However, in Conway and Norton’s paper, in the last paragraph of Section 9 which deals with moonshine for other groups, it was asked if there is “…a period three automorphism for the case Γ0(67)+”.  (The discriminant d = -67 is, of course, the younger brother of d = -163 in the family of Heegner numbers.)

III. More Pi Formulas?

In addition to the very interesting fact that the moonshine functions span a linear space of dimension 163, one question we can ask is, since there are so many of them, it is possible to use other McKay-Thompson series TpA in Ramanujan-type pi formulas?  Ramanujan’s four kinds of pi formulas are of the form,

where hp is one of various products of three Pochhammer symbols (their factorial equivalents are given in this article) and C is either j(τ), r2A(τ), r3A(τ), r4A(τ) or equivalently, the McKay-Thompson series T1A, T2A, T3A, T4A with an appropriate constant term.  It is tempting to speculate that, perhaps using different Pochhammer symbols and an appropriate constant term, one can use, say, T7A as well.

In fact, J. Guillera has extended Ramanujan’s results and found pi formulas of form,

where gp is one of various products of  five Pochhammer symbols, but apparently the denominators are not values of any McKay-Thompson series.  See “A Compilation of Ramanujan-Type Formulas for 1/πm” for m > 1 for a list.

IV. A Continued Fraction Using eπ√163

First, given the curious prime-generating polynomial,

P(n) = 4n2+163

which is prime for n = {0 to 19}.  Solve for P(n) = 0, hence n = (1/2)√-163, plug this into r4A(n) given in the Introduction and we find that it is an algebraic number,

r4A((1/2)√-163) = x24

where x ≈ 5.31863…  is the real root of the simple cubic equation x3-6x2+4x-2 = 0Second, given the Ramanujan constant R,

R = eπ√163 ≈ 6403203 + 743.9999999999992…

and we see that R and x have the beautiful relationships,

and,

where q = -1/R = -1/eπ√163.  The root x has the exact value, let τ = (1+√-163)/2,

where η(τ) is the Dedekind eta function which partly explains the two relationships between R and x.  While this continued fraction is not unique to d = 163 (the general form is by E. Heine, 1846), I thought of including it in this article since R is a well-known constant. The significance of the integer 24 will be further elaborated in other articles.  Furthermore, it is also the case that,

eπ√43 ≈ 123(92-1)3 + 743.999…

eπ√67 ≈ 123(212-1)3 + 743.99999…

eπ√163 ≈ 123(2312-1)3 + 743.999999999999…

The reason for the squares within the cubes is a certain Eisenstein series.  But that’s another story ...

-- End --

© July 2011, Tito Piezas III

You can email author at tpiezas@gmail.com.