# Klein's j-invariant and Ford circles

Klein's j-invariant has structure which seems to resemble Ford circles:

The latter show up all over number theory (continued fractions, Rademacher's expansion for p(n), etc.)

Can someone explain the connection?

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What is the diagram showing? – Gerry Myerson Nov 21 '12 at 23:39
don't all modular forms have that symmetry? – sperners lemma Nov 21 '12 at 23:42
Technically, modular forms transform with power-laws under the modular group, and only those of weight one are invariant. (With $\tau\mapsto\tau+1$ and $\tau\mapsto-\tau^{-1}$ as the generators of $\Gamma$, a modular form of weight $k$ transforms as $f(-1/\tau)=\tau^k f(\tau)$.) So modular forms of nonzero weight don't exactly have the same symmetry. – anon Nov 22 '12 at 2:24
[^Error: meant to say weight zero in the first sentence.] – anon Nov 22 '12 at 3:10
+1 for the pretty picture :-) – David Loeffler Nov 22 '12 at 9:36

Ford circles are the orbit of a horocycle under the action of the modular group $\Gamma$ on the upper half plane $\frak h$ (see e.g. this entry of SBS). Modular forms of weight zero (of which the $j$-invariant is an instance) are fully $\mathrm{SL}_2(\Bbb Z)$-invariant. So the periods of $j$ (representing the quotient $H/\Gamma$) tile the hyperbolic plane (of which $\frak h$ is a model) according to the same symmetry as the Ford circle tiling.
The Ford circles are also a diagram of $PSL_2 \mathbb Z.$ A fairly good discussion is in The Sensual Quadratic Form by John H. Conway. The circles are horocycles instead of geodesics. Compare http://en.wikipedia.org/wiki/Modular_group#Tessellation_of_the_hyperbolic_plane
Just to begin the part I really know, if you have a binary quadratic form $$f(x,y) = a x^2 + b x y + c y^2,$$ which we abbreviate as $\langle a,b,c \rangle,$ the traditional question is the possible primitively represented values of $f,$ that is $f(p,q)$ with $\gcd(p,q) = 1.$ However, this is really no different from finding the equivalence class of the form. Equivalence is probably best illustrated with the Hessian matrix of second partial derivatives $$H =\left( \begin{array}{cc} 2a & b \\ b & 2c \end{array} \right).$$
Take a matrix in $SL_2 \mathbb Z$ and multiply with that matrix on the right of H and its transpose on the left, as in $$\left( \begin{array}{cc} \alpha & \gamma \\ \beta & \delta \end{array} \right) \left( \begin{array}{cc} 2a & b \\ b & 2c \end{array} \right) \left( \begin{array}{cc} \alpha & \beta \\ \gamma & \delta \end{array} \right) \; = \; \left( \begin{array}{cc} 2A & B \\ B & 2C \end{array} \right).$$ The result is the Hessian matrix of a new form $\langle A,B,C \rangle.$ The relationship to primitively represented values is that $f(\alpha, \gamma) = A.$