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"""Fourier coefficients of monstrous McKay–Thompson series -- numberdb.org/T652.
Run with SageMath and the attached seeds.json and class-data.json:
sage -pip install numberdb
sage -python generate.py # verify all stored entries
sage -python generate.py --fill # write entries to the existing draft
The --fill mode needs NUMBERDB_API_KEY. It never makes the draft public.
Borcherds, Monstrous moonshine and monstrous Lie superalgebras (1992),
§9, equation (9.1), determines the coefficients from n=1,2,3,5 and the
squared-class series. The seed values were transcribed from Conway–Norton,
Table 4, printed p. 334; H_0 in that source is not a Fourier constant term.
All divisions by two below require exact divisibility. Class power maps are
frozen from GAP CTblLib 1.3.11. Identity coefficients are computed internally
from E4^3/Delta-744, but excluded from enumeration because T79 covers them.
See argument-selection.md for both discrete indices and the finite bounds.
"""
import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from functools import lru_cache
from pathlib import Path
import json
import os
import sys
HERE = Path(__file__).resolve().parent
SEEDS = json.loads((HERE / 'seeds.json').read_text())
CLASS_DATA = json.loads((HERE / 'class-data.json').read_text())
CLASSES = tuple(g for g, order in zip(CLASS_DATA['class_names'],
CLASS_DATA['class_orders'])
if 1 < order <= 10)
SQUARE = CLASS_DATA['power_maps']['2']
assert len(CLASSES) == 32
assert set(SQUARE) == set(CLASSES) | {'1A'}
assert set(SQUARE.values()) <= set(SQUARE)
def product(a, b, degree):
return [sum((a[j] * b[n-j] for j in range(max(0,n-len(b)+1),
min(n,len(a)-1)+1)), ZZ(0)) for n in range(degree+1)]
def identity_series(nmax):
"""Exact E4^3 / product(1-q^m)^24, shifted by q^-1 and minus 744."""
degree = nmax + 1
e4 = [ZZ(1)] + [ZZ(240)*sum(ZZ(d)**3 for d in range(1,n+1)
if n % d == 0) for n in range(1,degree+1)]
numerator = product(product(e4,e4,degree),e4,degree)
denominator = [ZZ(1)] + [ZZ(0)]*degree
for m in range(1,degree+1):
for _ in range(24):
for k in range(degree,m-1,-1):
denominator[k] -= denominator[k-m]
quotient = []
for n in range(degree+1):
quotient.append(numerator[n] - sum(denominator[j]*quotient[n-j]
for j in range(1,n+1)))
assert quotient[0] == 1 and quotient[1] == 744
return {n:quotient[n+1] for n in range(1,nmax+1)}
def half(value):
value = ZZ(value)
assert value % 2 == 0
return value // 2
@lru_cache(maxsize=None)
def coefficients(nmax=30):
assert nmax >= 5
data = {g:{int(n):ZZ(v) for n,v in SEEDS[g].items()} for g in CLASSES}
data['1A'] = identity_series(nmax)
for n in range(4,nmax+1):
if n == 5:
continue
k, r = divmod(n,4)
for g in CLASSES:
a = data[g]
b = data[SQUARE[g]]
if r == 0:
value = a[2*k+1] + half(a[k]**2-b[k])
value += sum(a[j]*a[2*k-j] for j in range(1,k))
elif r == 1:
value = a[2*k+3]-a[2]*a[2*k]+half(a[2*k]**2+b[2*k])
value += half(a[k+1]**2-b[k+1])
value += sum(a[j]*a[2*k-j+2] for j in range(1,k+1))
value += sum(b[j]*a[4*k-4*j] for j in range(1,k))
value += sum((-1)**j*a[j]*a[4*k-j] for j in range(1,2*k))
elif r == 2:
value = a[2*k+2]+sum(a[j]*a[2*k-j+1] for j in range(1,k+1))
else:
value = a[2*k+4]-a[2]*a[2*k+1]-half(a[2*k+1]**2-b[2*k+1])
value += sum(a[j]*a[2*k-j+3] for j in range(1,k+2))
value += sum(b[j]*a[4*k-4*j+2] for j in range(1,k+1))
value += sum((-1)**j*a[j]*a[4*k-j+2] for j in range(1,2*k+1))
a[n] = ZZ(value)
return data
class McKayThompsonCoefficients(numberdb.Generator):
table = 'T652'
parameters = ('g','n')
type = 'Z'
rigour = 'exact'
files = ('generate.py','seeds.json','class-data.json','argument-selection.md',
'checks.py','check.py','published-controls.json','gap_export.py')
def enumerate(self, nmax=30):
for g in CLASSES:
for n in range(1,nmax+1):
yield {'g':g, 'n':n}
def value(self, params, digits):
n = int(params['n'])
return coefficients(max(30,n))[params['g']][n]
if __name__ == '__main__':
if os.environ.get('NUMBERDB_KEY_FROM_STDIN') == '1':
os.environ['NUMBERDB_API_KEY'] = sys.stdin.read().strip()
generator = McKayThompsonCoefficients()
if '--fill' in sys.argv or os.environ.get('NUMBERDB_PUBLISH') == '1':
print(generator.publish(message='Fill 960 signed exact coefficients after independent checks'))
else:
report = generator.verify(sample=None)
print(report)
sys.exit(0 if report.ok else 1)