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), and every modular form in this series is defined by how it behaves under this group. This part shows that the group is built from a shift and an inversion with only two relations between them, and that only two points of ℍ, up to symmetry, are fixed by anything other than ±I.
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. Type entries or pick a preset. The grey box is the word in S and T found by the Euclidean algorithm below.
The special linear group SL₂(ℤ) consists of all 2×2 matrices with integer entries and determinant 1:
It is an infinite, discrete subgroup of SL₂(ℝ), acting on ℍ by the Möbius transformations of Part 1:
The imaginary part of the image is positive, so the action maps ℍ to itself, and large c pushes points toward the real axis:
The matrices γ and −γ give the same transformation, since the signs cancel. The group that acts faithfully on ℍ is 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, reflects across the imaginary axis and fixes i.
Every γ in SL₂(ℤ) is a product of S, T and T⁻¹, up to sign, by the Euclidean algorithm on the first column (a, c). While c ≠ 0, take the integer n nearest to a/c. Multiplying on the left by T⁻ⁿ replaces a with a − nc, which is smaller than |c|, and then S⁻¹ swaps the column to (c, −(a − nc)), so the lower-left entry has strictly shrunk. A determinant-1 integer matrix with c = 0 is ±Tᵐ. Collecting the steps gives
For [[5,2],[7,3]] Figure 1 finds γ = TST⁴ST²ST in three rounds. The exponents 1, 4, 2 are the partial quotients of the continued fraction 5/7 = 1 − 1/(4 − 1/2), because each round is one division step on a/c = γ · ∞.
The generators are not independent. Squaring S gives −I, and so does cubing ST:
So in PSL₂(ℤ) the element S has order 2 and U = ST has order 3, and every other identity between words follows from these two:
The symbol ∗ is the free product: each element is exactly one word that alternates S with U or U⁻¹. In these letters T = SU and T⁻¹ = U⁻¹S.
Uniqueness has a picture. The images of the arc of the unit circle from i to ρ = e2πi/3 form a tree: S turns the arc about i and U turns it about ρ, so two arcs meet at each image of i and three at each image of ρ. A word is a walk along the tree, one arc per letter, and a reduced word never doubles back; in a tree only one such walk joins two arcs.
Figure 2. Images of the arc from i to ρ, in Part 1's disk: two meet at each image of i (green), three at each image of ρ (purple). Each button appends letters on the right; the red path is the reduced word's walk to the shaded tile γ·F.
The arcs crowd toward the boundary but never pile up inside: SL₂(ℤ) is discrete, so the images of any point of ℍ stay a positive hyperbolic distance apart. 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. The exceptions are the vertices of the tree. S · i = −1/i = i, so the stabiliser of i is {I, S} ≅ ℤ/2, and S turns the plane by π about i. Since ST · τ = −1/(τ + 1) and ρ + 1 = eπi/3, ST · ρ = −e−πi/3 = ρ, so the stabiliser of ρ is {I, ST, (ST)²} ≅ ℤ/3, and ST turns the plane about ρ by 2π/3.
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 about the fixed point, and the coloured geodesics from the fixed point meet at 180° about i and 120° about ρ.
There are no other elliptic points up to equivalence. In Part 1's terms, γ ≠ ±I fixes a point of ℍ only when it is elliptic, |tr γ| < 2, so an integer matrix needs tr γ ∈ {−1, 0, 1}. Trace 0 makes γ conjugate in SL₂(ℤ) to ±S, fixing a point in the orbit of i; trace ±1 makes it conjugate in PSL₂(ℤ) to (ST)±1, fixing a point in the orbit of ρ. Figure 4 checks this on every elliptic matrix with small entries.
Figure 4. Fixed points of every elliptic matrix, up to sign, with entries of size at most N, over the tiles γ · F. Trace 0 (green circles) and trace ±1 (purple triangles) sit on edge midpoints and corners of tiles; the readout reduces each one into F.
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 fractional correction terms when Part 5 counts modular forms.