The graphic nature of Gaussian periods

Recent work has shown that the study of supercharacters on abelian groups provides a natural framework within which to study 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, our approach reveals that these classical objects display dazzling visual patterns of great complexity and remarkable subtlety.


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][2][3]. Recent work has shown that the study of supercharacters on Abelian groups provides a natural framework within 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 σ X " ÿ χPX χp0qχ is constant on each K P K. The characters σ X 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 χ x pyq " ep xy n q for x in Z{nZ, where epθq " expp2πiθq. For a fixed subgroup A of pZ{nZqˆ, let K denote the partition of Z{nZ arising from the action a¨x " ax of A. The action a¨χ x " χ a´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 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 cyclotomy. For p " kd`1, Gauss defined the d-nomial periods η j " ř d´1 "0 ζ g k `j p , Partially supported by National Science Foundation Grant DMS-1001614.
where ζ p " ep 1 p q and g denotes a primitive root modulo p [4,6,14]. Clearly η j runs over the same values as σ X pyq when y ‰ 0, |A| " d, and X " A1 is the A-orbit of 1. For composite moduli, the functions σ X attain values which are generalizations of Gaussian periods of the type considered by Kummer and others (see [11]). If σ X pyq and σ X py 1 q differ in color, then y ı y 1 pmod mq, where m is a certain fixed proper divisor of n. Coloring the points σ X pyq according to residue classes of y can reveal hidden structure.
When visualized as subsets of the complex plane, the images of these supercharacters 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. 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 overwhelming. To some degree, these circumstances force us to focus our initial efforts on documenting the notable features that appear and on explaining their number-theoretic origins. One such theorem is the following. Theorem 1.1. Suppose that q is an odd prime power and that σ X is a cyclic supercharacter of Z{qZ. If |X| " d is prime, then the image of σ X is bounded by the m-cusped hypocycloid parametrized by θ Þ Ñ pd´1qe iθ`eipd´1qθ .
In fact, for a fixed prime m, as the modulus q " 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.  The preceding theorem is itself a special case of a much more general theorem (Theorem 6.3) which relates the asymptotic behavior of cyclic supercharacter plots to the mapping properties of certain multivariate Laurent polynomials, regarded as complex-valued functions on a suitable, high-dimensional torus.

Multiplicativity and nesting plots
Our first order of business is to determine when and in what manner the image of one cyclic supercharacter plot can appear in another. Certain cyclic supercharacters have a naturally multiplicative structure. When combined with Proposition 2.4 and the discussion in Section 6, the following result provides a complete picture of the boundaries of these supercharacters. Following the introduction, we let X " Ar denote the orbit of r in Z{nZ under the action of a cyclic unit subgroup A.
Theorem 2.1. Let σ X be a cyclic supercharacter of Z{nZ, writing n " ś k j"1 p aj j in standard form and X " xωyr. For each j, let ψ j : Z{nZ Ñ Z{p aj j Z be the natural homomorphism, let x j be the multiplicative inverse of n{p aj j pmod p aj j q, and write X j " xψ j pωqyx j ψ j prq. If the orbit sizes |X j | are pairwise coprime, then Proof. We prove the theorem for n " p 1 p 2 a product of distinct primes; the general argument is similar. Let ψ " pψ 1 , ψ 2 q be the ring isomorphism given by the Chinese Remainder Theorem, and let d " |X|, d 1 " |ψ 1 pXq| and d 2 " |ψ 2 pXq|. We have As a consequence of the next result, we observe all possible graphical behavior, up to scaling, by restricting our attention to cases where r " 1 (i.e., where X " A as sets). We present it without proof.
Proposition 2.2. Let r belong to Z{nZ, and suppose that pr, nq " n d for some positive divisor d of n, so that ξ " rd n is a unit modulo n. Also let ψ : Z{nZ Ñ Z{dZ be the natural homomorphism.
(ii) The image in (i), when scaled by |A| |ψ d pAq| , is a subset of the image of σ Aξ .
where X " x319yr. Each image nests in Figure 3(f), as per Proposition 2.2(ii). See Figure 1 for a brief discussion of colorization.
Example 2.3. Let n " 62160 " 2 4¨3¨5¨7¨3 7. Each plot in Figure 3 displays the image of a different cyclic supercharacter σ X , where X " x319yr. If d " r{pn, rq, then Proposition 2.2(i) says that each image equals that of a cyclic supercharacter σ X 1 of Z{dZ, where X 1 " xψ d p319qy1. Proposition 2.2(ii) says that each nests in the image in Figure 3(f).
In part because of Theorem 2.1, we are especially interested in cyclic supercharacters with prime power moduli. The following result implies that the image of any cyclic supercharacter of Z{p a Z is essentially a scaled copy of one whose boundary is given by Theorem 6.3.
Proposition 2.4. Let p be an odd prime, a ą b nonnegative integers, and ψ the natural homomorphism from Z{p a Z to Z{p a´b Z. If σ X is a cyclic supercharacter of Z{p a´b Z, where X " A1 with p b |X| and p a´b " 1 pmod |ϕpXq|q, then

Symmetries
We say that a cyclic supercharacter σ X : Z{nZ Ñ C has k-fold dihedral symmetry if its image is invariant under the natural action of the dihedral group of order 2k. In other words, σ X has k-fold dihedral symmetry if its image is invariant under complex conjugation and rotation by 2π{k about the origin. If X is the orbit of r, where pr, nq " n d for some odd divisor d of n, then σ X is generally asymmetric about the imaginary axis, as evidenced by Figure 4.
Odd values of n{pr, nq can produce asymmetric images. See Figure  1 for a brief discussion of colorization.

It follows that
Since the function e is periodic with period 1, we have In other words, the image of σ X is invariant under counterclockwise rotation by 2πξ{k about the origin. If mξ " 1 pmod kq, then the graph is also invariant under counterclockwise rotation by m¨2πξ{k " 2π{k. Dihedral symmetry follows, since for all y in Z{nZ, the the image of σ X contains both σ X pyq and σ X pyq " σ X p´yq. Consider the cyclic supercharacter σ X1 , whose graph appears in Figure 4(b). We have p20485, 4608q " p5¨17¨241, 2 9¨32 q " 1, so Theorem 3.1 guarantees that σ X1 has 1-fold dihedral symmetry. It is visibly apparent that σ X has only the trivial rotational symmetry. Figures 5(a) to 5(f) display the graphs of σ Xm in the cases m ‰ 1. For each such m, the graph of σ Xm contains a scaled copy of σ X1 by Theorem 2.2 and has m-fold dihedral symmetry by Theorem 3.1, since p20485m, 4608q " m. It is evident from the associated figures that m is maximal in each case, in the sense that σ Xm having k-fold dihedral symmetry implies k ď m.

Real and imaginary supercharacters
The images of some cyclic supercharacters are subsets of the real axis. Many others are subsets of the union of the real and imaginary axes. In this section, we establish sufficient conditions for each situation to occur and provide explicit evaluations in certain cases. Let σ X be a cyclic supercharacter of Z{nZ, where X " Ar. If A contains´1, then it is immediate from (1) that σ X is real-valued.
Example 4.1. Let X be the orbit of 3 under the action of x164y on Z{855Z. Since 164 3 "´1 pmod nq, it follows that σ X is real-valued, as suggested by Figure 6(a).
We turn our attention to cyclic supercharacters whose values, if not real, are purely imaginary (see Figure 7). To this end, we introduce the following notation. Let k be a positive divisor of n, and suppose that A " xj 0 n{k´1y , for some 1 ď j 0 ă k. ( In this situation, we have Figure 5. Graphs of cyclic supercharacters σ X of Z{nZ, where X " x4609y1. By taking multiples of n, we produce dihedrally symmetric images containing the one in Figure 4(b), each rotated copy of which is colored differently.
so that every element of A has either the form jn k`1 or jn k´1 , where 0 ď j ă k. In this situation, we write for some subsets J`and J´of t0, 1, . . . , k´1u. The condition (3) is vacuous if k " n. However, if k ă n and j 0 ą 1 (i.e., if A is nontrivial), then it follows that p´1q |A| " 1 pmod n k q, whence |A| is even. In particular, this implies |J`| " |J´|. The subsets J`and J´are not necessarily disjoint. For instance, if A " x´1y " t´1, 1u, then (3) holds where k " 1 and J`" J´" t0u. In general, J`must contain 0, since A must contain 1. The following result is typical of those obtainable by imposing restrictions on J`and J´. (i) If r is even, then the image of σ X is a subset of the real axis.
(ii) If r is odd, then σ X pyq is real whenever y is even and purely imaginary whenever y is odd.
Proof. Each x in X has the form pjn{k`1qr or ppk{2´jq n{k`1q r. If y " 2m for some integer m, then for every summand epxy{nq in the definition of σ X pyq having the form e p2mpjn{k`1qr{nq, there is one of the form e p2mpn{2´jn{k`1qr{nq, its complex conjugate. From this we deduce that σ X pyq is real whenever y is even. If y " 2m`1, then for every summand of the form e pp2m`1q pjn{k`1q r{nq, there is one of the form e pp2m`1qpn{2´jn{k`1qr{nq. If r is odd, then the latter is the former reflected across the imaginary axis, in which case σ X pyq is purely imaginary. If r is even, then the latter is the complex conjugate of the former, in which case σ X pyq is real.

Ellipses
Discretized ellipses appear frequently in the graphs of cyclic supercharacters. These, in turn, form primitive elements whence more complicated supercharacter plots emerge. In order to proceed, we recall the definition of a Gauss sum. Suppose that m and k are integers with k ą 0. If χ is a Dirichlet character modulo k, then the Gauss sum associated with χ is given by If p is prime, the quadratic Gauss sum gpm; pq over Z{pZ is given by gpm; pq " gpm, χq, where χpaq "´a p¯i s the Legendre symbol of a and p. That is, We require the following well-known result [4, Thm. 1.5.2].
Proposition 5.2. Suppose that p|n and p " 1 pmod 4q is prime. Let Q p " tm P Z{pZ :ˆm p˙" 1u denote the set of distinct nonzero quadratic residues modulo p. If (3) holds where J`" taq`b : q P Q p u and J´" tcq´b : q P Q p u (4) for integers a, b, c coprime to p with´a p¯"´´c p¯, then σ X pyq belongs to the real interval r1´p, p´1s whenever p|y, and otherwise belongs to the ellipse described by the equation pRe zq 2`p Im zq 2 {p " 1.
Proof. For all y in Z{nZ, we have σ X pyq " where θ y " pbn`pqy pn . If p|y, then epθ y q " ep y n q and ep a 2 y p q " ep c 2 y p q " 1, so σ X pyq " pp´1q 2ˆe´y n¯`e´y n¯˙" pp´1q cos 2πy n .
p 2´e pθ y q´epθ y q¯´cos 2πθ y "˘iˆy p˙? p sin 2πθ y´c os 2πθ y , where (5) follows from Lemma 5.1.

Asymptotic behavior
We now turn our attention to an entirely different matter, namely the asymptotic behavior of cyclic supercharacter plots. To this end we begin by recalling several definitions and results concerning uniform distribution modulo 1. The discrepancy of a finite subset S of r0, 1q m is the quantity where the supremum runs over all boxes B " ra 1 , b 1 qˆ¨¨¨ˆra m , b m q and µ denotes m-dimensional Lebesgue measure. We say that a sequence S n of finite subsets of r0, 1q d is uniformly distributed if lim nÑ8 DpS n q " 0. If S n is a sequence of finite subsets in R m , we say that S n is uniformly distributed mod 1 if the corresponding sequence of sets ptx 1 u, tx 2 u, . . . , tx d uq : px 1 , x 2 , . . . , x m q P S n ( is uniformly distributed in r0, 1q m . Here txu denotes the fractional part x´txu of a real number x. The following fundamental result is due to H. Weyl [15]. In the following, we suppose that q " p a is a nonzero power of an odd prime and that |X| " d is a divisor of p´1. Let ω q denote a primitive dth root of unity modulo q and let S q " " q p1, ω q , ω 2 q , . . . , ω ϕpdq´1 q q : " 0, 1, . . . , q´1 where ϕ denotes the Euler totient function. The following lemma of Myerson, whose proof we have adapted to suit our notation, can be found in [12,Thm. 12].
Lemma 6.2. The sets S q for q " 1 pmod dq are uniformly distributed modulo 1.
Let r " q{pq, f pω q qq, and observe that Having fixed d and v, we claim that the sum above is nonzero for only finitely many q " 1 pmod dq. Letting Φ d denote the dth cyclotomic polynomial, recall that deg Φ d " ϕpdq and that Φ d is the minimal polynomial of any primitive dth root of unity. Clearly the gcd of f ptq and Φ d ptq as polynomials in Qrts is in Z. Thus there exist aptq and bptq in Zrts so that aptqΦ d ptq`bptqf ptq " n for some integer n. Passing to Z{qZ and letting t " ω q , we find that bpω q qf pω q q " n pmod pq. This means that q|f pω q q implies q|n, which can occur for only finitely many prime powers q. Putting this all together, we find that for all v in Z ϕpdq the following holds: By Weyl's Criterion, it follows that the sets S q are uniformly distributed mod 1 as q " 1 pmod dq tends to infinity. Theorem 6.3. Let σ X be a cyclic supercharacter of Z{qZ, where q " p a is a nonzero power of an odd prime. If X " A1 and |X| " d divides p´1, then the image of σ X is contained in the image of the function g : r0, 1q ϕpdq Ñ C defined by gpz 1 , z 2 , . . . , z ϕpdq q " where the integers b k,j are given by For a fixed d, as q becomes large, the image of σ X fills out the image of g, in the sense that, given ą 0, there exists some q " 1 pmod dq such that if σ X : Z{qZ Ñ C is a cyclic supercharacter with |X| " d, then every open ball of radius ą 0 in the image of g has nonempty intersection with the image of σ X .
Proof. Let ω q be a primitive dth root of unity modulo q, so that A " xω q y in from which it follows that the image of σ X is contained in the image of the function g : T ϕpdq Ñ C defined by (6). The density claim now follows immediately from Lemma 6.2.
In combination with Propositions 2.2 and 2.4, the preceding theorem characterizes the boundary curves of cyclic supercharacters with prime power moduli. If d is even, then X is closed under negation, so σ X is real. If d " k a where k is an odd prime, then g : T ϕpk a q Ñ C is given by gpz 1 , z 2 ,¨¨¨, z ϕpdq q " ϕpdq ÿ j"1 z j`k a´1 ÿ j"1 k´2 ź "0 z´1 j` k a´1 .
A particularly concrete manifestation of our result is Theorem 1.1, whose proof we present below. Recall that a hypocycloid is a planar curve obtained by tracing the path of a distinguished point on a small circle as it rolls within a larger circle. Rolling a circle of integral radius λ within a circle of integral radius κ, where κ ą λ, yields the parametrization θ Þ Ñ pκ´λqe iθ`λ e p1´κ{λqiθ of the hypocycloid centered at the origin that contains the point κ and has precisely κ cusps. . Cyclic supercharacters σ X of Z{qZ, where X " A1, whose graphs fill out |X|-hypocycloids.
The image of the function g : T d´1 Ñ C defined above is the filled hypocycloid corresponding to the parameters κ " d and λ " 1, as observed in [10, §3].