Finite geometry

Abstract: A point in a finite projective plane PG(2, pn), may be denoted by the symbol (Xl, X2, X3), where the coordinates x1, X2, X3 are marks of a Galois field of order pn, GF(pn). The symbol (0, 0, 0) is excluded, and if k is a non-zero mark of the GF(pn), the symbols (X1, X2, X3) and (kxl, kx2, kx3) are to be thought of as the same point. The totality of points whose coordinates satisfy the equation ulxl+u2x2+U3x3 = 0, where u1, U2, u3 are marks of the GF(pn), not all zero, is called a line. The plane then consists of p2n +pn + 1 = q points and q lines; each line contains pn+1 points.t A finite projective plane, PG(2, pn), defined in this way is Pascalian and Desarguesian; it exists for every prime p and positive integer n, and there is only one such PG(2, pn) for a given p and n (VB, p. 247, VY, p. 151). Let Ao be a point of a given PG(2, pn), and let C be a collineation of the points of the plane. (A collineation is a 1-1 transformation carrying points into points and lines into lines.) Suppose C carries Ao into Al, A1 into A2,... , Ak into Ao; or, denoting the product C C by C2, C. C2 by C3, etc., we have C(Ao) =A1, C2(Ao) =A2, . . , Ck(A o) =A o. If k is the smallest positive integer for which C k(A o) =Ao, we call k the period of C with respect to the point A o. If the period of a collineation C with respect to a point Ao is q (=p2n+pn+l), then the period of C with respect to any point in the plane is q, and in this case we will call C simply a collineation of period q. We prove in the first theorem that there is always at least one collineation of period q, and from it we derive some results of interest in finite geometry and number theory. Let

Abstract: Introduction Cast of characters Part I: 1. Congruences and the quotient ring of the integers mod n 1.2 The discrete Fourier transform on the finite circle 1.3 Graphs of Z/nZ, adjacency operators, eigenvalues 1.4 Four questions about Cayley graphs 1.5 Finite Euclidean graphs and three questions about their spectra 1.6 Random walks on Cayley graphs 1.7 Applications in geometry and analysis 1.8 The quadratic reciprocity law 1.9 The fast Fourier transform 1.10 The DFT on finite Abelian groups - finite tori 1.11 Error-correcting codes 1.12 The Poisson sum formula on a finite Abelian group 1.13 Some applications in chemistry and physics 1.14 The uncertainty principle Part II. Introduction 2.1 Fourier transform and representations of finite groups 2.2 Induced representations 2.3 The finite ax + b group 2.4 Heisenberg group 2.5 Finite symmetric spaces - finite upper half planes Hq 2.6 Special functions on Hq - K-Bessel and spherical 2.7 The general linear group GL(2, Fq) 2.8. Selberg's trace formula and isospectral non-isomorphic graphs 2.9 The trace formula on finite upper half planes 2.10 The trace formula for a tree and Ihara's zeta function.

Abstract: Planar functions were introduced by Dembowski and Ostrom [4] to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3 ^e for every e ≥ 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.