On page 227 of Symmetry and the Monster, Mark Ronan tells the story of Conway and Norton computing the number of independent _mini j-functions_ (McKay-Thompson series) arising from the Moonshine module. There are 194 distinct characters of the monster (btw. see the background picture for the first page of the character table as given in the Atlas), but some of them give the same series reducing the number of series to 171. But, these are not all linearly independent. Mark Ronan writes :
“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 special property of 163 in number theory has intruiging consequences, among which is the fact that
is very close to being a whole number.”
The corresponding footnote is a bit cryptic and doesn’t explain this near miss integer.
“This special feature also yields a fact, first noticed by Euler, that the formula
gives prime numbers for all values of x between 1 and 40. The connection with 163 is that the solution to
involves the square root of -163.”
So, what is really going on?
The _modular j-function_ has a power series expansion in
and classifies complex elliptic curves upto isomorphism, or equivalently, two-dimensional integral lattices upto a complex scaling factor. A source of two-dimensional integral lattices is given by the rings of integers
Leopold Kronecker discovered in 1857 the remarkable fact that the modular j-function detects the class number of
The function-value
Special instances of this theorem were already known. For example, the Gaussian integers
and because
Charles Hermite noticed in 1859 this curious numerical consequence of Kronecker’s theorem. For, if one takes
So, all but the first two terms in the series expansion of
Combining this information with the Gauss-computed value of
whence the observed curious approaximation of
What about other near misses which follow from Kronecker’s result? Unfortunately there are only nine imaginary quadratic extension
and of course the near misses will be worse for smaller values of D. For example for the next two largest values one calculates
Reference :
John Stillwell, Modular Miracles, The American Mathematical Monthly, 108 (2001) 70-76
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