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?
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?
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 $f(\alpha, \gamma) = A.$