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The Graphic Nature of Gaussian Periods William Duke Claremont Colleges Scholarship @ Claremont Pomona Faculty Publications and Research Pomona Faculty Scholarship 1-1-2013 The Graphic Nature of Gaussian Periods William Duke Stephan Ramon Garcia Pomona College Bob Lutz '13 Pomona College Recommended Citation Duke, William; Garcia, Stephan Ramon; and Lutz, Bob '13, "The Graphic Nature of Gaussian Periods" (2013). Pomona Faculty Publications and Research. 232. http://scholarship.claremont.edu/pomona_fac_pub/232 This Article - preprint is brought to you for free and open access by the Pomona Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in Pomona Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. THE GRAPHIC NATURE OF GAUSSIAN PERIODS WILLIAM DUKE, STEPHAN RAMON GARCIA, AND BOB LUTZ Abstract. Recent work has shown that the study of supercharacters on abelian groups provides a natural framework with which to study the proper- ties of certain exponential sums of interest in number theory. Our aim here is to initiate the study of Gaussian periods from this novel perspective. Among other things, this approach reveals that these classical objects display a daz- zling array of visual patterns of great complexity and remarkable subtlety. 1. Introduction The theory of supercharacters, which generalizes classical character theory, was recently introduced in an axiomatic fashion by P. Diaconis and I.M. Isaacs [7], extending the seminal work of C. André [1–3]. Recent work has shown that the study of supercharacters on abelian groups provides a natural framework with which to study the properties of certain exponential sums of interest in number theory [5,9] (see also [8]). In particular, Gaussian periods, Ramanujan sums, Kloosterman sums, and Heilbronn sums can be realized in this way (see Table 1). Our aim here is to initiate the study of Gaussian periods from this novel perspective. Among other things, this approach reveals that these classical objects display a dazzling array of visual patterns of great complexity and remarkable subtlety (see Figure 1). Let G be a finite group with identity 0, K a partition of G, and X a partition of the set IrrpGq of irreducible characters of G. The ordered pair pX ; Kq is called a supercharacter theory for G if t0u P K, |X | “ |K|, and for each X P X , the (generalized) character σX “ χp0qχ χPX ¸ is constant on each K P K. The characters σX are called supercharacters of G and the elements of K are called superclasses. Let G “ Z{nZ and recall that the irreducible characters of Z{nZ are the functions xy χxpyq “ ep n q for x in Z{nZ, where epθq “ expp2πiθq. For a fixed subgroup A of arXiv:1212.6825v2 [math.NT] 22 May 2013 pZ{nZqˆ, let K denote the partition of Z{nZ arising from the action a¨x “ ax of A. Similarly, the action a¨χx “ χa´1x of A on the irreducible characters of Z{nZ yields a compatible partition X . The reader can verify that pX ; Kq is a supercharacter theory on Z{nZ and that the corresponding supercharacters are given by xy σ pyq “ e ; (1) X n xPX ¸ ´ ¯ where X is an orbit in Z{nZ under the action of a subgroup A of pZ{nZqˆ. When n “ p is an odd prime, (1) is a Gaussian period, a central object in the theory of d´1 gk``j cyclotomy. For p “ kd ` 1, Gauss defined the d-nomial periods ηj “ `“0 ζp , Partially supported by National Science Foundation Grant DMS-1001614. ř 1 2 W. DUKE, S. R. GARCIA, AND BOB LUTZ where ζp “ expp2πi{pq and g denotes a primitive root modulo p [4,6,14]. Clearly ηj runs over the same values as σX pyq when y ‰ 0, |A| “ d, and X “ A is the A-orbit of 1. For composite moduli, the functions σX attain values which are generalizations of Gaussian periods of the type considered by Kummer and others (see [11]). (a) n “ 52059, A “ x766y (b) n “ 91205, A “ x2337y (c) n “ 70091, A “ x3447y (d) n “ 91205, A “ x39626y (e) n “ 91205, A “ x1322y (f) n “ 95095, A “ x626y (g) n “ 82677, A “ x8147y (h) n “ 70091, A “ x21792y (i) n “ 51319, A “ x430y Figure 1. Each subfigure is the image of σX : Z{nZ Ñ C where X is the orbit of r “ 1 under the action of A ¤ pZ{nZqˆ on Z{nZ. When visualized as subsets of the complex plane, the images of these superchar- acters exhibit a surprisingly diverse range of features (see Figure 1). The main purpose of this paper is to initiate the investigation of these plots, focusing our attention on the case where A “ xay is a cyclic subgroup of pZ{nZqˆ. We refer to supercharacers which arise in this manner as cyclic supercharacters. THE GRAPHIC NATURE OF GAUSSIAN PERIODS 3 Name Expression G A d´1 gk``j Gauss ηj “ e Z{pZ nonzero kth powers mod p p `“0 ¸ ˆ ˙ n jx ˆ Ramanujan cnpxq “ e Z{nZ pZ{nZq n j“1 ˆ ˙ pj;n¸q“1 p´1 a` ` b` 2 u 0 ˆ Kloosterman Kppa; bq “ e pZ{pZq 1 : u P pZ{pZq p 0 u´ `“0 ˜ ¸ ¸ "„ * p´1 p a` 2 2 Heilbronn Hppaq “ e Z{p Z nonzero pth powers mod p p2 `“0 ¸ ˆ ˙ Table 1. Gaussian periods, Ramanujan sums, Kloosterman sums, and Heilbronn sums appear as supercharacters arising from the action of a subgroup A of Aut G for a suitable abelian group G. Here p denotes an odd prime number. The sheer diversity of patterns displayed by cyclic supercharacters is overwhelm- ing. To some degree, these circumstances force us to focus our initial efforts on doc- umenting the notable features that appear and on explaining their number-theoretic origins. One such theorem is the following. Theorem 1.1. Suppose that p is an odd prime and that σX is a cyclic supercharac- ter on Z{pZ. If |X| “ d is prime, then the image of σX is bounded by the d-cusped hypocycloid parametrized by θ ÞÑ pd ´ 1qeiθ ` eipd´1qθ. In fact, for a fixed prime d, as the modulus p ” 1 pmod dq tends to infinity the corresponding supercharacter images become dense in the filled hypocycloid in a sense that will be made precise in Section 6. (a) p “ 2791, A “ x800y (b) p “ 27011, A “ x9360y (c) p “ 202231, A “ x61576y Figure 2. Graphs of cyclic supercharacters σX of Z{pZ, where X “ A1, showing the density of hypocycloids as p Ñ 8. The preceding theorem is itself a special case of a more general theorem (The- orem 6.3) which relates the asymptotic behavior of cyclic supercharacter plots to 4 W. DUKE, S. R. GARCIA, AND BOB LUTZ the mapping properties of certain multivariate Laurent polynomials, regarded as complex-valued functions on a suitable, high-dimensional torus. 2. Nesting plots Our first order of business is determining when the image of one cyclic super- character plot is contained in another. Following the introduction, we let X “ Ar denote the orbit of an element r of Z{nZ under the action of a cyclic subgroup ˆ A “ xay of pZ{nZq . For each positive divisor d of n, let d : Z{nZ Ñ Z{dZ be the ring homomorphism that maps each element of Z{nZ to its residue modulo d, so ˆ that dpAq “ x dpaqy ¤ pZ{dZq acts on Z{dZ by multiplication. As a consequence of the following result, we can observe all possible graphical behavior by restricting our attention to the case r “ 1. n Proposition 2.1. Let r belong to Z{nZ, and suppose that pr; nq “ d for some rd positive divisor d of n, so that ξ “ n is a unit modulo n. (i) The images of σAr, σApr;nq, and σ dpAq1 are equal. (ii) The image in (i) is a scaled subset of the image of σAξ. Proof. Label the supercharacters in the statement of the theorem in order of men- tion as σXj , where j “ 1; 2; 3; 4. For each j, let Cj be a subset of A (resp., dpAq) such that |Cj| “ |Xj| and if Xj is the orbit of x, then Xj “ tcx : c P Cju. Since cξn{d ” c1ξn{d pmod nq if and only if cn{d ” c1n{d pmod nq, we may assume that 1 ´1 C1 “ C2. For any y in Z{nZ, letting y “ ξ y gives xy1 cξy1 cy σ py1q “ e “ e “ e “ σ pyq; X1 n d d X2 xPX cPC cPC ¸1 ˆ ˙ ¸1 ˆ ˙ ¸2 ´ ¯ 1 2 so im σX2 Ă im σX1 . Letting y “ ξy instead gives σX2 py q “ σX1 pyq, proving the reverse containment. Since cn{d ” c1n{d pmod nq if and only if c ” c1 pmod dq, we may assume that dpC2q “ C3 and |C2| “ |C3|. We then have cy cy σ pyq “ e “ e “ pσ ˝ qpyq; X2 d d X3 d cPC cPC ¸2 ´ ¯ ¸3 ´ ¯ since e is periodic with period 1. Hence im σX2 “ im σX3 , concluding the proof of (i). Since cr ” c1r pmod nq if and only if cξ ” c1ξ pmod dq, we may assume that 1 C1 “ dpC4q. For any y in Z{nZ, letting y “ yn{d gives 1 cy |C4| cy |A| σX4 py q “ e “ e “ σX1 pyq; d | dpC4q| d | dpAq| cPC c C ¸4 ´ ¯ P ¸dp 4q ´ ¯ which concludes the proof of (ii). Example 2.2. Let n “ 62160 “ 24 ¨ 3 ¨ 5 ¨ 7 ¨ 37. Each plot in Figure 3 displays n the image of a different cyclic supercharacter σX , where X is the orbit of d under the action of x319y on Z{nZ for some given positive divisor d of n.
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