Schmidt arrangements

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Schmidt arrangements

This picture from a paper by my colleague Katherine Stange shows the Schmidt arrangement of the Gaussian integers. It is formed by taking the orbit of the real line under the action of a particular group of symmetries of the extended complex plane. This gives a graphical depiction of some of the algebraic properties of quadratic imaginary fields and their rings of integers.

The complex numbers are numbers of the form z=a+bi, where a and b are real numbers and i is (one of) the square roots of –1. The complex number a+bi can be identified with the point (a,b) in the plane. The Gaussian rationals, Q[i], are the complex numbers z for which a and b are both rational, and the Gaussian integers, Z[i], are the complex numbers z for which a and b are both integers. 

The Gaussian integers form a ring, which means that the sum, difference, or product of two Gaussian integers gives another such number. In addition to these properties, the Gaussian rationals also have the feature that if z is a nonzero Gaussian rational, then 1/z is also a Gaussian rational. This means that the Gaussian rationals form a field.

The relationship between Z[i] and Q[i] is like the relationship between Z (the integers) and Q (the rationals). The Gaussian rationals Q[i] are the field of fractions of the Gaussian integers Z[i], meaning that Q[i] is the smallest field that contains Z[i]. Conversely, Z[i] is the ring of integers of Q[i], which means that it consists of all the elements of Q[i] that are roots of monic polynomials with integer coefficients. More specifically, every Gaussian integer is a root of a polynomial of the form x^2+bx+c, where b and c are integers. For example, the Gaussian integer 2+3i is a root of the quadratic polynomial x^2–4x+13.

The extended complex plane, also known as the Riemann sphere, is a way to add an extra point, called infinity, to the usual complex numbers. With this identification, straight lines and circles in the complex plane both correspond to circles on the Riemann sphere, the only difference being that the straight lines in the plane correspond to circles containing the point at infinity, and circles in the plane correspond to circles on the Riemann sphere that avoid the point at infinity. These straight lines and circles are sometimes known by the umbrella term of circlines.

The connection with the picture comes via the Bianchi groups. These are constructed from the group SL_2(Z[i]), which consists of the set of 2 by 2 matrices with determinant 1 and entries from Z[i], under the operation of matrix multiplication. The Bianchi group, PSL_2(Z[i]), is obtained from SL_2(Z[i]) by identifying each matrix with its negative (in technical terms: taking the quotient of SL_2(Z[i]) by the scalar matrices).

Suppose that A is an element of the Bianchi group SL_2(Z[i]), represented by a matrix with first row [a c] and second row [b d]; in this case, the “determinant 1” condition means precisely that ad–bc=1. The element A corresponds to a transformation of the extended complex plane sending z to (az+c)/(bz+d), and this transformation sends circlines to circlines. The Schmidt arrangement results from applying every possible Bianchi group element to the circline given by the horizontal axis y=0.

The red and blue arrangement in the picture is the result of performing this procedure with PGL_2(Z[i]). This group is twice as large as PSL_2(Z[i]), and the condition ad–bc=1 is replaced by the weaker condition that ad–bc should come from the set {1, i, –1, –i}. The Schmidt arrangement consists of the red curves in the picture, which shows the square region of the complex plane between x=0, x=1, y=0, and y=1. The groups of symmetries here are infinite, and the very small circles have been omitted from the picture. 

The paper from which this picture is taken is Visualising the arithmetic of quadratic imaginary fields ( by Katherine E. Stange. The paper treats a more general case than the Gaussian rationals, namely the case of the field Q(√d), where d is a negative integer with no square factors other than 1. The paper proves that the curvatures of the circles are all integer multiples of √(—Δ) (where Δ is the discriminant), and that the circles are in bijection with certain ideal classes in orders of the ring of integers. Furthermore, the ring of integers is a Euclidean domain if and only if this arrangement of circles is connected; these conditions are satisfied in the case of the Gaussian integers.

Relevant links

Riemann sphere (extended complex plane):

A great illustration by +Henry Segerman of the correspondence between the Riemann sphere and the plane:

Quadratic integer:

Quadratic field:

#mathematics #scienceeveryday   #spnetwork  arXiv:1410.0417

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