Modular Forms

Delta = eta^24

Modular forms are functions on the upper half-plane with extraordinary symmetry. The most famous is the discriminant Delta(tau), whose coefficients define the Ramanujan tau function. Reducing tau(p) modulo 23 classifies all primes into 3 classes, one per ideal class of Q(sqrt(-23)) where h(-23) = 3. The primes {2, 3, 5, 7, 11, 13} all sit in either the -1 class or the 0 class.

The Tau Classification

For any prime p (except 23), Ramanujan's tau function satisfies: 23 divides tau(p) if and only if p is a quadratic non-residue mod 23. The Legendre symbol sorts the primes:

Prime ptau(p)tau(p) mod 23Class
2-24-1QR (inner)
3252-1QR (inner)
548300NR (outer)
7-167440NR (outer)
115346120NR (outer)
13-577738-1QR (inner)

Inner primes {2, 3, 13} are quadratic residues mod 23. Outer primes {5, 7, 11} are non-residues. The Ramanujan congruence primes -- tau(n) has famous congruences at 5, 7, and 11 -- are exactly the outer class. All three hit tau mod 23 = 0.

Three-Value Theorem

Three-Value Classification
For primes p (not 23), tau(p) mod 23 takes exactly 3 values: {0, 2, -1}. The class number h(-23) = 3. Three classes: (1) Non-residue primes: tau = 0. (2) Residue primes on non-principal form 2x^2+xy+3y^2: tau = -1. (3) Residue primes on principal form x^2+xy+6y^2: tau = 2. All six primes {2,3,5,7,11,13} sit in class 1 or 2. The form 2x^2+xy+3y^2 represents 2 at (1,0), 3 at (0,1), 13 at (2,1). The first principal-form prime is 59. For all n: tau(n) mod 23 takes 4 values {0, 1, 2, -1}; 1 appears only at composites (tau(1) = 1).

The (2, 1, 3) Form

Why 3 values? The imaginary quadratic field Q(sqrt(-23)) has class number h(-23) = 3. Its two quadratic forms of discriminant -23:

Non-principal
2x^2 + xy + 3y^2
Coefficients (2, 1, 3). Represents 2 at (1,0), 3 at (0,1), 13 at (2,1).
Principal
x^2 + xy + 6y^2
Coefficients (1, 1, 6). First prime: 59, then 101, 167, 173...
23
2*11 + 1
Cunningham image of 11: c(11) = 23.
h(-23) = 3
Class group = Z/3Z
3 controls the splitting of primes in this field.

The Galois representation of Delta mod 23 is dihedral, induced from a Hecke character of Q(sqrt(-23)). For inert primes (NR): trace = 0. For split primes: trace depends on which of 3 ideal classes the prime above p falls into. Non-principal class: trace = -1. Principal class: trace = 2. The primes {2, 3, 13} all land in the non-principal class.

The Number 1728

The j-invariant normalization uses 1728 = 12^3. Delta = (E4^3 - E6^2) / 1728.

1728
12^3 = (4 * 3)^3
j-invariant normalization.
12
lcm(1, 2, 3, 4) = 12
Weight of Delta. Carmichael lambda of Z/210.
24
2 * 12
Leech lattice dimension. Exponent of eta in Delta = eta^24.

The Number 691

Ramanujan proved: tau(n) = sigma_11(n) mod 691. This is the deepest congruence of the discriminant modular form.

691 mod 210
61
691 = 3 * 210 + 61.
691 mod 23
1
Identity in Z/23.
Prime index
691 = 125th prime
125 = 5^3.

The Ring of Modular Forms

All modular forms are polynomials in just two generators: E4 (weight 4) and E6 (weight 6). Every Eisenstein weight is even:

FormWeightFactorizationNote
E442^2Generator 1
E662 * 3Generator 2
E882^3E8 Lie algebra rank
E10102 * 5
E12122^2 * 3Weight of Delta
E14142 * 7

Cusp Form Duality

Cusp forms = modular forms that vanish at the cusps. The space S_k has dimension:

k = 12
dim = 1
Unique cusp form: Delta.
k = 24
dim = 2
Two cusp forms at Leech lattice dimension.
k < 12
dim = 0
No cusp forms below weight 12.

The first cusp form appears at weight 12 = Carmichael lambda of Z/210. The same number 12 that bounds the partition smooth zone (see Eta Bridge page).

The Partition-Eta Thread

The generating function for partitions is 1/eta(tau). The eta function's 24th power = Delta. 24 = 8 * 3 links three structures: the Cunningham chain c(n) = 2n+1, the Leech lattice (24 dimensions), and the modular form (eta^24 = Delta). The partition function p(n) is {2,3,5,7,11}-smooth for n <= 12. The unique cusp form has weight 12. Same number, same threshold.

Explore: Tau Classification

Enter a prime p. See its class: QR inner (tau = -1 or 2) or NR outer (tau = 0). The primes {2, 3, 13} are inner. {5, 7, 11} are outer. 59 is the first principal-form prime.

Prime p:

Contrast Table

Modular formsAbstract automorphic functions. 1728 and 691 appear from internal logic1728 = 12^3. 691 mod 210 = 61. Both factor through the ring's primes.Tau classificationCoefficients of a q-series, studied for analytic propertiestau(p) mod 23 sorts all primes into 3 classes (h(-23) = 3). Form coefficients = (2, 1, 3).Ramanujan congruencestau(n) has congruences at primes 5, 7, 11 (discovered empirically)Exactly {5, 7, 11} = the outer class mod 23 (non-residues).Why 1728?Normalization constant for j-invariant12^3. Weight 12 = Carmichael lambda of Z/210.Cusp form thresholdFirst cusp form at weight 1212 = Carmichael lambda of Z/210 = partition smooth zone boundary.

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