On the Dimension of the Space of Harmonic Functions on a Discrete Torus

Abstract

Let d(n) denote the corank of I + A over the field with two elements, where A is the adjacency matrix of the discrete torus C_n × C_n , and I is the identity matrix. We shall prove that d(2n) = 2d(n) and d(2 ^r + 1) = d(2 ^r − 1) + 4. For the proof of the latter result, we use an elliptic curve. Our motivation for this study is the “lights out” puzzle.

Keywords:

• Lights out puzzle
• graph Laplacian
• discrete torus
• elliptic curve
• Chebyshev-Dickson polynomials