Part 2 of 14. Modular forms are defined by how they transform under the modular group SL₂(ℤ), which is built from just two moves: a shift and an inversion.
Part 1 ended with SL₂(ℝ) acting on ℍ by isometries. The matrices with integer entries form the modular group SL₂(ℤ), a discrete group of symmetries. Every modular form in this series is defined by how it behaves under this group, so this part gets to know it. Every element is a product of two simple ones, a shift and an inversion, and only two points of ℍ, up to symmetry, are fixed by anything other than ±I. Those two points come back in Part 5 as the fractional terms in the count of modular forms.
Figure 1 applies one matrix γ to a point τ. Drag the blue point or type integer entries. The readout writes γ as a product of the generators S and T. The section on the generators below shows why such a product always exists.
Figure 1. The Möbius action of γ ∈ SL₂(ℤ) on a draggable point τ (blue); the image γ·τ is red, and the dashed circle is |τ| = 1. The grey line is the word in S and T found by the Euclidean algorithm described below, recomputed whenever the entries change.
The special linear group SL₂(ℤ) consists of all 2×2 matrices with integer entries and determinant 1:
This is an infinite, discrete subgroup of SL₂(ℝ). Its elements act on the upper half-plane ℍ by the same Möbius transformation that defines isometries of the Poincaré metric:
The imaginary part of the image shows that the action maps ℍ to itself:
The numerator is positive because ad − bc = 1 and Im τ > 0, and the denominator |cτ + d|² is positive, so Im(γ · τ) > 0. Large c pushes points toward the real axis.
The matrices γ and −γ give the same Möbius transformation, since the signs cancel. The group that acts faithfully on ℍ is therefore the quotient PSL₂(ℤ) = SL₂(ℤ)/{±I}.
SL₂(ℤ) is generated by two elements:
T shifts every point one unit to the right. S sends τ to −1/τ, which inverts the modulus and reflects across the imaginary axis. It swaps the inside of the unit circle with the outside and fixes i, since −1/i = i.
Every γ in SL₂(ℤ) is a product of S, T and T⁻¹, up to sign. The proof is the Euclidean algorithm run on the first column (a, c). If c ≠ 0, pick the integer n nearest to a/c. Multiplying on the left by T⁻ⁿ replaces a with a − nc, which is smaller than |c| in absolute value. Multiplying on the left by S⁻¹ then swaps the column into (c, −(a − nc)), so the lower-left entry has strictly shrunk. Repeat until c = 0. A determinant-1 integer matrix with c = 0 has a = d = ±1, so it is ±Tᵐ. Collecting the steps gives
Figure 1 runs this loop on whatever matrix you enter. For the preset [[2,1],[1,1]] it finds γ = T²ST. For [[5,2],[7,3]] it takes three rounds and returns γ = TST⁴ST²ST. The exponents 1, 4, 2 are the partial quotients of the continued fraction 5/7 = 1 − 1/(4 − 1/2), because each round of the loop is one division step on the fraction a/c = γ · ∞.
The generators are not independent. Squaring S gives −I, and so does cubing ST:
Since −I acts trivially on ℍ, in PSL₂(ℤ) the element S has order 2 and ST has order 3. Every other identity between words in S and T follows from these two:
The symbol ∗ is the free product. An element of ℤ/2 ∗ ℤ/3 is an alternating word in a letter of order 2 and a letter of order 3, and two such words are equal only if they are identical. It has trivial centre and contains free subgroups of every finite rank.
In Figure 2 each click applies one more generator to τ, the purple trail follows the images, and the matrix under the buttons is the running product. Click S twice, or S then T three times over, and the matrix returns to −I while the red point lands back on the blue one.
Figure 2. Word builder. Each button applies a generator to the current image, so the product is written right to left: pressing S then T builds TS. The shaded region is the fundamental domain F of the next explainer. The status line reduces the product back to Euclidean normal form, which collapses to ±I whenever the word is a consequence of S² = (ST)³ = −I.
Because SL₂(ℤ) is discrete, the images of τ never pile up at a point of ℍ. They crowd toward the real axis in the Euclidean picture, but in the hyperbolic metric any two distinct images are at least as far apart as τ is from its nearest image. Two points in the same orbit are called SL₂(ℤ)-equivalent, and Part 3 chooses one representative from every orbit.
For most τ the only element of PSL₂(ℤ) that fixes τ is the identity. There are two exceptions, and they come from the relations.
The first is i. S · i = −1/i = i, so the stabiliser of i in PSL₂(ℤ) is {I, S} ≅ ℤ/2. Near i, S acts as a half-turn: it rotates the plane by π about i.
The second is ρ = e2πi/3 = (−1 + i√3)/2, a primitive cube root of unity. Since ST · τ = −1/(τ + 1) and ρ + 1 = eπi/3, we get ST · ρ = −e−πi/3 = ρ. The stabiliser of ρ is {I, ST, (ST)²} ≅ ℤ/3, and ST rotates the plane about ρ by 2π/3.
Drag the probe point in Figure 3. Its images under the stabiliser stay on one hyperbolic circle around the fixed point, and the geodesic rays to them meet at equal angles, 180° at i and 120° at ρ.
Figure 3. The stabiliser of an elliptic point acting on a draggable probe (filled dot). Hollow dots are its images, the solid circle is the hyperbolic circle through the probe centred at the fixed point, and the coloured arcs are the geodesics from the fixed point. The status line gives the hyperbolic distances and the angles between consecutive geodesics at the fixed point.
There are no other elliptic points up to equivalence. A matrix γ ∈ SL₂(ℝ), γ ≠ ±I, fixes a point of ℍ only when |tr γ| < 2. For an integer matrix that leaves tr γ ∈ {−1, 0, 1}. Trace 0 means γ² = −I, and every such γ is conjugate in SL₂(ℤ) to ±S, so it fixes a point in the orbit of i. Trace ±1 means γ has order 3 in PSL₂(ℤ) and is conjugate there to (ST)±1, so it fixes a point in the orbit of ρ. Every elliptic point of SL₂(ℤ) is therefore equivalent to i or to ρ. The corner ρ + 1 = −ρ̄ of the fundamental domain is T · ρ, so it lies in the orbit of ρ.
In the quotient SL₂(ℤ)\ℍ these two orbits become cone points. A neighbourhood of i folds in half, and a neighbourhood of ρ folds into thirds, giving cone angles π and 2π/3. The fundamental domain of Part 3 shows this folding, and the cone points reappear as the fractional correction terms when Part 5 counts modular forms.