PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 5, May 2015, Pages 1849–1863 S 0002-9939(2015)12322-2 Article electronically published on January 8, 2015

THE GRAPHIC NATURE OF GAUSSIAN PERIODS

WILLIAM DUKE, STEPHAN RAMON GARCIA, AND BOB LUTZ

(Communicated by Ken Ono)

Abstract. 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 pat- terns 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 [6], extending the seminal work of C. Andr´e [1, 2]. 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 [4, 8] (see also [7] and 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,andX a partition of the set IrrpGq of irreducible characters of G. The ordered pair pX , Kq is called a supercharacter theory for G if t0uPK, |X |“|K|,andforeachX P X , the function ÿ σX “ χp0qχ χPX

is constant on each K P K. The functions 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 p q“ p xy q Z{ Z p q“ p q χx y e n for x in n ,wheree θ exp 2πiθ . For a fixed subgroup A of pZ{nZqˆ,letK denote the partition of Z{nZ arising from the action a ¨ x “ ax of A. 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 ÿ ´xy ¯ (1) σ pyq“ e , X n xPX

Received by the editors July 26, 2013. 2010 Mathematics Subject Classification. Primary 11L05, 11L99, 11T22, 11T23, 11T24. The first author was partially supported by National Science Foundation Grants DMS-10- 01527, DMS-1001614, and DMS-1265973.

c 2015 American Mathematical Society Reverts to public domain 28 years from publication 1849

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where X is an orbit in Z{nZ under the action of A.Whenn “ p is an odd prime, (1) is a Gaussian period, a central object in the theory of cyclotomy. For p “ kd ` 1, “ řd´1 gk`j “ p 1 q Gauss defined the d-nomial periods ηj “0 ζp ,whereζp e p and g denotes a primitive root modulo p [3, 5]. Clearly ηj runs over the same values as σX pyq when y ‰ 0, |A|“d,andX “ A1istheA-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 [10]).

Figure 1. Each subfigure is the image of σX : Z{nZ Ñ C, where X is the orbit of 1 under the action of a cyclic subgroup ˆ 1 1 A of pZ{nZq .IfσX pyq and σX py q differ in color, then y ı y pmod mq,wherem is a fixed divisor of n.

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

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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.

Name Expression G A d´1 ÿ ˆ gk`j ˙ Gauss η “ e Z{pZ nonzero kth powers mod p j p “0

n ÿ ˆ jx˙ Ramanujan c pxq“ e Z{nZ pZ{nZqˆ n n j“1 pj,nq“1 p´1 ÿ ˆa ` b ˙ "„u 0 j * Kloosterman K pa, bq“ e pZ{pZq2 : u PpZ{pZqˆ p p 0 u´1 “0

p´1 ÿ ˆ ap ˙ Heilbronn H paq“ e Z{p2Z nonzero pth powers mod p2 p p2 “0

attention on the case where A “xωy is a cyclic subgroup of pZ{nZqˆ. We refer to supercharacters which arise in this manner as cyclic supercharacters. 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.

a Theorem 1.1. Suppose that q “ p is an odd prime power and that σX is a cyclic supercharacter of Z{qZ.If|X|“d is prime and d ‰ p, then the image of σX is bounded by the d-cusped hypocycloid parametrized by θ ÞÑpd ´ 1qeiθ ` 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 (see Figures 2 and 9).

Figure 2. Graphs of cyclic supercharacters σX : Z{pZ Ñ C where X “ A1.

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The preceding result 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 suitable high-dimensional tori.

2. 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 these supercharacters. Following the introduction, we let X “ Ar denote the orbit of r in Z{nZ under the action of a cyclic subgroup A of pZ{nZqˆ. The following is a simple consequence of the Chinese Remainder Theorem.

Theorem 2.1. Suppose that σxωyr is a cyclic supercharacter on Z{mnZ,where pm, nq“1,andletρ : Z{mnZ Ñ Z{mZ ˆ Z{nZ be the natural isomorphism. If ρpωq“pωm,ωnq, ρprq“prm,rnq,anda, b P Z with mb ` na “ 1, then for all y P Z{mnZ we have p q“ p q p q σxωyr y σxωmyrm ay σxωnyrn by . The following easy result tells us that we observe all possible graphical behavior, up to scaling, by studying cases where r “ 1 (i.e., where X “ A as sets). P Z{ Z p q“ n Proposition 2.2. Let r n , and suppose that r, n d for some positive “ rd Z{ Z Ñ Z{ Z divisor d of n, so that ξ n is a unit modulo n.Alsoletψd : n d be reduction modulo d.

(i) The images of σAr and σApr,nq are equal.

(ii) The images of σψdpAq1 and σAξ are equal. (iii) The image in (i), when scaled by |A| , is a subset of the image in (ii). |ψdpAq|

Example 2.3. Let n “ 62160 “ 24 ¨ 3 ¨ 5 ¨ 7 ¨ 37.

Each plot in Figure 3 displays the image of a different cyclic supercharacter σX , where X “x319yr.Ifd “ r{pn, rq, then Proposition 2.2(i) says that each im- 1 age equals that of a cyclic supercharacter σX1 of Z{dZ,whereX “xψdp319qy1. Proposition 2.2(ii) says that each nests in the image in Figure 3(F).

Because of Theorem 2.1, we are mostly interested in prime power moduli. The following result implies that the image of any cyclic supercharacter on Z{paZ is 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{paZ to Z{pa´bZ.Ifσ is a cyclic supercharacter ˇ X of Z{pa´bZ,whereX “ A1 with pbˇ|X| and pa´b ” 1 pmod |ψpXq|q,then

a b a´b σX pZ{p Zq“t0uYp σϕpXqpZ{p Zq.

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Figure 3. Graphs of cyclic supercharacters σX of Z{62160Z, where X “x319yr. Each image nests in Figure 3(F), as per Propo- sition 2.2(ii). See Figure 1 for a brief discussion of colorization.

Proof. Let k|pp ´ 1q.If|X|“kpb,thenA “ ψ´1pA1q,whereA1 is the unique subgroup of pZ{pa´bZqˆ of order k.LetX1 “ A11 (i.e., X1 “ ψpXq), so that

X “tx ` jpa´b : x P X1,j“ 0, 1,...,pb ´ 1u.

We have

pb´1 ÿ ÿ ˆpx ` jpa´bqy ˙ σ pyq“ e X pa xPX1 j“0 pb´1 ÿ ˆjy˙ ÿ ˆxy ˙ “ e e pb pa j“0 xPX1 # b b p σ 1 pψpyqq, if p |y “ X 0, else.



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3. Symmetries

We say that a cyclic supercharacter σX : 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, σX 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 p q“ n r,where r, n d for some odd divisor d of n,thenσX is generally asymmetric about the imaginary axis, as evidenced by Figure 4.

Figure 4. Graphs of σX : Z{nZ Ñ C,whereX “ Ar,fixingr “ 1 (close inspection reveals that (E) enjoys no rotational symmetry).

Proposition 3.1. If σX is a cyclic supercharacter of Z{nZ,whereX “xωyr,then p ´ n q σX has ω 1, pr,nq -fold dihedral symmetry.

Proof. Let d “ n{pr, nq.Ifk “pω ´ 1,dq,thenω, and hence every element of xωy, has the form jk ` 1. Since r “ ξn{d for some unit ξ,eachx in X has the form pξn{dqpjk ` 1q.Ify1 “ y ` d{k,theny1 ´ y ´ d{k ” 0 pmod nq,inwhichcase

ξn ˆ d ˙ pjk ` 1q y1 ´ y ´ ” 0 pmod nq. d k

It follows that ˆξn ξn˙ pjk ` 1q `y1 ´ y˘ ´ ” 0 pmod nq, d k

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whence ξn ξn ξn pjk ` 1qy1 ” pjk ` 1qy ` pjk ` 1qpmod nq d d k ξn ξn ” pjk ` 1qy ` pmod nq. d k Since the function e is periodic with period 1, we have ÿ ˆxy1 ˙ ÿ ˆxy ` ξn{k ˙ ˆ ξ ˙ ÿ ´xy ¯ e “ e “ e e . n n k n xPX xPX xPX

In other words, the image of σX 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 image of σX contains both σX pyq and σX pyq“σX p´yq. 

Figure 5. Graphs of cyclic supercharacters σX of Z{nZ,where X “x4609y1. One can produce dihedrally symmetric images con- taining the one in Figure 3(B), each rotated copy of which is colored differently.

Example 3.2. For m “ 1, 2, 3, 4, 6, 8, 12, let Xm denote the orbit of 1 under the Z Z action of x4609y on {p20485mq . Consider the cyclic supercharacter σX1 ,whose graph appears in Figure 3(B). We have p20485, 4608q“p5 ¨ 17 ¨ 241, 29 ¨ 32q“1,

so Proposition 3.1 guarantees that σX1 has 1-fold dihedral symmetry. It is visibly apparent that σX has no rotational symmetry.

Figures 5(A) to 5(F) display the graphs of σXm in the cases m ‰ 1. For each

such m, the graph of σXm contains a scaled copy of σX1 by Proposition 2.2 and

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has m-fold dihedral symmetry by Proposition 3.1, since p20485m, 4608q“m.It is evident from the associated figures that m is maximal in each case, in the sense

that σXm having k-fold dihedral symmetry implies k ď m. 4. 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 σX be a cyclic supercharacter of Z{nZ,where X “ Ar.IfA contains ´1, then it is immediate from (1) that σX is real-valued. Example 4.1. Let X be the orbit of 3 under the action of x164y on Z{855Z.Since 3 164 ”´1 pmod nq, it follows that σX is real-valued, as suggested by Figure 6(A). “x´y “ ‰ n “t´ u Example 4.2. If A 1 and X Ar where r 2 ,thenX r, r and σX pyq“2cosp2πry{nq. Figure 6(B) illustrates this situation.

Figure 6. Graphs of cyclic supercharacters σX of Z{nZ,where X “ A1. Each σX is real-valued, since each A contains ´1.

Figure 7. Graphs of cyclic supercharacters σX : Z{nZ Ñ C whose values are either real or purely imaginary.

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

(2) A “xj0n{k ´ 1y , for some 1 ď j0 ă k. In this situation, we write

(3) A “tjn{k ` 1:j P J`uYtjn{k ´ 1:j P J´u

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 j0 ą 1 (i.e., if p´ q|A| ” p n q | | A is nontrivial), then it follows that 1 1 mod k , whence A is even. In particular, this implies |J`|“|J´|. The subsets J` and J´ are not necessarily

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disjoint. For instance, if A “x´1y“t´1, 1u, then (3) holds where k “ 1and J` “ J´ “t0u. In general, J` must contain 0, since A must contain 1. The following result is typical of those obtained by imposing restrictions on J` and J´.

Proposition 4.3. Let σX be a cyclic supercharacter of Z{nZ,whereX “ Ar,and “ k ´ suppose that (3) holds, where k is even and J´ 2 J`. (i) If r is even, then the image of σX is a subset of the real axis.

(ii) If r is odd, then σX 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.Ify “ 2m for some integer m, then for every summand epxy{nq in the definition of σX 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 σX pyq is real whenever y is even. If y “ 2m ` 1, then for every summand of the form e pp2m ` 1qpjn{k ` 1q r{nq,there is one of the form e pp2m ` 1qpn{2 ´ jn{k ` 1qr{nq.Ifr is odd, then the latter is the former reflected across the imaginary axis, in which case σX pyq is purely imaginary. If r is even, then the latter is the conjugate of the former, whence σX pyq is real. 

Example 4.4. In the case of Figure 7(A), we have n “ 912, r “ 1, k “ 38, j0 “ 3,

J` “t0, 2, 12, 16, 20, 22, 24, 26, 32u, and J´ “t3, 7, 17, 19, 25, 31, 33, 35, 37u, so the hypotheses of Proposition 4.3(ii) hold.

An explicit evaluation of σX is available if J` Y J´ “t0, 1,...,k ´ 1u.The following result, presented without proof, treats this situation (see Figure 7(B)).

Proposition 4.5. Suppose that k ą 2 is even, and that (3) holds where J` is the set of all even residues modulo k and J´ is the set of all odd residues. If X is the orbit of a unit r under the action of A on Z{nZ,then $ k cos 2πry if k|y, &’ n p q“ 2πry ” k p q σX y ik sin n if y 2 mod k , %’0 otherwise.

5. Ellipses Discretized ellipses appear frequently in the graphs of cyclic supercharacters. These, in turn, form primitive elements from which more complicated superchar- acter 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 k ÿ ˆ m˙ Gpm, χq“ χp qe . k “1 If p is prime, the quadratic Gauss sum gpm; pq over Z{pZ is given by gpm; pq“ ´a¯ gpm, χq,whereχpaq“ p istheLegendresymbolofa and p.Thatis,

k´1 ÿ ˆm 2 ˙ gpm; pq“ e . p “0 We require the following well-known result [3, Thm. 1.5.2].

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Lemma 5.1. If p ” 1 pmod 4q is prime and pm, pq“1,then ˆm˙? gpm; pq“ p. p Proposition 5.2. Suppose that p|n and p ” 1mod4is prime. If (3) holds where “t 2 ` “ p´1 u “t 2 ´ “ p´1 u (4) J` a b : 1, 2,..., 2 and J´ c b : x 1, 2,..., 2 ´ ¯ ´ ¯ Z a c for a, b, c P coprime to p with p “´p ,thenσX pyq belongs to the real interval r1 ´ p, p ´ 1s whenever p|y, and otherwise belongs to the ellipse described by the equation pRe zq2 `pIm zq2{p “ 1. Proof. For all y in Z{nZ,wehave ÿ ´xy ¯ σ pyq“ e X n xPA ˜p jn ` q ¸ ˜p jn ´ q ¸ ÿ p 1 y ÿ p 1 y “ e ` e n n jPJ` jPJ´ pp´1q{2 pp´1q{2 ÿ ˆpa 2 ` bqy y ˙ ÿ ˆpc 2 ´ bqy y ˙ “ e ` ` e ´ p n p n “1 “1 pp´1q{2 pp´1q{2 ˆby y ˙ ÿ ˆa 2y ˙ ˆ by y ˙ ÿ ˆc 2y ˙ “ e ` e ` e ´ ´ e p n p p n p “1 “1 pp´1q{2 pp´1q{2 ÿ ˆa 2y ˙ ÿ ˆc 2y ˙ “ epθ q e ` epθ q e , y p y p “1 “1 2 2 “ pbn`pqy | p q“ p y q p a y q“ p c y q“ where θy pn .Ifp y,thene θy e n and e p e p 1, so pp ´ 1q ˆ ´ y ¯ ´ y ¯˙ 2πy σ pyq“ e ` e “pp ´ 1q cos . X 2 n n n If not, then pp, yq“1, so e pθ qpgpay; pq´1q`epθ qpgpcy; pq´1q σ pyq“ y y X 2

epθyqgpay; pq`epθyqgpcy; pq “ ´ cos 2πθy ? 2 p ˆˆay˙ ˆcy˙ ˙ (5) “ epθ q` epθ q ´ cos 2πθ 2 p y p y y ? ˆy˙ p ´ ¯ “˘ epθ q´epθ q ´ cos 2πθ p 2 y y y ˆy˙? “˘i p sin 2πθ ´ cos 2πθ , p y y where (5) follows from Lemma 5.1. 

Example 5.3. Let n “ 1535 “ 5 ¨ 307 and consider the cyclic supercharacter σx613y1 on Z{nZ. In this situation, illustrated by Figure 8(A),the hypotheses of Proposition 5.2 hold with pa, b, cq“p2, 2, 1q. Figure 8(B) illustrates the situation

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Figure 8. Graphs of cyclic supercharacters σX of Z{nZ,where X “ A1. Propositions 2.2, 3.1 and 5.2 can be used to produce supercharacters whose images feature elliptical patterns.

pa, b, cq“p1, 3, 2q, while Figure 8(C) illustrates p1, 1, 2q. The remainder of Fig- ure 8 demonstrates the effect of using Propositions 2.2, 3.1 and 5.2 to produce supercharacters whose images feature ellipses.

6. 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 discrepency of a finite subset S of r0, 1qm is the quantity ˇ ˇ ˇ |B X S| ˇ DpSq“sup ˇ ´ μpBqˇ , B ˇ |S| ˇ

where the supremum runs over all boxes B “ra1,b1qˆ¨¨¨ˆram,bmq and μ denotes m-dimensional Lebesgue measure. We say that a sequence Sn of finite subsets of d r0, 1q is uniformly distributed if limnÑ8 DpSnq“0. If Sn is a sequence of finite m subsets in R ,wesaythatSn is uniformly distributed mod 1 if the corresponding  ( sequence of sets ptx1u, tx2u,...,txduq : px1,x2,...,xmqPSn is uniformly dis- tributed in r0, 1qm.Heretxu denotes the fractional part x ´ txu of a real number x. The following fundamental result is due to H. Weyl [13]. m Lemma 6.1. A sequence of finite sets Sn in R is uniformly distributed modulo 1 if and only if 1 ÿ lim epu ¨ vq“0 nÑ8 |Sn| uPSn

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for each v in Zm. In the following, we suppose that q “ pa is a nonzero power of an odd prime and that |X|“d is a divisor of p ´ 1. Let ωq denote a primitive dth root of unity modulo q and let "  * S “ p1,ω ,ω2,...,ωϕpdq´1q :  “ 0, 1,...,q´ 1 Ďr0, 1qϕpdq q q q q q 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 [11, Thm. 12].

Lemma 6.2. The sets Sq for q ” 1 pmod dq are uniformly distributed modulo 1.

ϕpdq Proof. Fix a nonzero vector v “pa0,a1,...,aϕpdq´1q in Z and let

ϕpdq´1 fptq“a0 ` a1t `¨¨¨`aϕpdq´1t .

Let r “ q{pq, fpωqqq, and observe that q´1 ˆ ˙ ÿ ÿ fpωqq epu ¨ vq“ e q uPSq “0 q{r´1 pm`1qr´1 ˆ ˙ ÿ ÿ fpωqq “ e r m“0 “mr r´1 ˆ ˙ q ÿ fpωqq “ e r r “0 # q if q|fpω q, “ q 0else. Having fixed d and v, we claim that the sum above is nonzero for only finitely many q ” 1 pmod dq. Letting Φd denote the dth cyclotomic polynomial, recall that deg Φd “ ϕpdq and that Φd is the minimal polynomial of any primitive dth root of unity. Clearly the gcd of fptq and Φdptq as polynomials in Qrts is in Z.Thus there exist aptq and bptq in Zrts so that aptqΦdptq`bptqfptq“n for some integer n. Passing to Z{qZ and letting t “ ωq, we find that bpωqqfpωqq”n pmod pq.This means that q|fpωqq implies that q|n, which can occur for only finitely many prime powers q. Putting this all together, we find that 1 ÿ lim epu ¨ vq“0 qÑ8 |Sq| q”1 pmod dq uPSq

ϕpdq holds for all v in Z . By Weyl’s Criterion, it follows that the sets Sq are uniformly distributed mod 1 as q ” 1 pmod dq tends to infinity. 

In what follows we let T denote the unit circle in the complex plane.

a Theorem 6.3. Let σX be a cyclic supercharacter of Z{qZ,whereq “ p is a nonzero power of an odd prime. If X “ A1 and |X|“d divides p ´ 1, then the ϕpdq image of σX is contained in the image of the function g : T Ñ C defined by

d´1 ϕpdq´1 ÿ ź bk,j (6) gpz1,z2,...,zϕpdqq“ zj`1 k“0 j“0

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where the integers bk,j are given by ϕpdq´1 k ÿ j (7) t ” bk,jt pmod Φdptqq. j“0

For a fixed d,asq becomes large, the image of σX fills out the image of g,inthe sense that, given  ą 0,thereexistssomeq ” 1 pmod dq such that if σX : 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 σX .

Proof. Let ωq be a primitive dth root of unity modulo q,sothatA “xωqy in pZ{ Zqˆ t p 1 q p ϕpdq´1 qu Z q . Recall that 1,e d ,...,e d is a -basis for the ring of integers of Qp p 1 qq “ ´ the cyclotomic field e d [12, Prop. 10.2]. For k 0, 1,...,d 1, the integers bk,j in the expression ϕpdq´1 ˆk ˙ ÿ ˆ j ˙ e “ b e d k,j d j“0 are determined by (7). In particular, it follows that ϕpdq´1 k ÿ j ωp ” bk,jωp pmod pq. j“0 We have ¨ ˛ ˆ ˙ d´1 ˜ k ¸ d´1 ϕpdq´1 j ÿ xy ÿ ωp ÿ ÿ ωp σX pyq“ e “ e “ e ˝ bk,j ‚, q p p xPX k“0 k“0 j“0

from which it follows that the image of σX is contained in the image of the function g : Tϕpdq Ñ C defined by (6). The density claim now follows from Lemma 6.2.  In combination with Propositions 2.2 and 2.4, the preceding theorem character- izes the boundary curves of cyclic supercharacters with prime power moduli. If d a is even, then X is closed under negation, so σX is real. If d “ p where p is an odd a prime, then g : Tϕpp q Ñ C is given by

ϕpdq pa´1 p´2 p ¨¨¨ q“ ÿ ` ÿ ź ´1 g z1,z2, ,zϕpdq zj zj`pa´1 . j“1 j“1 “0 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 which rolls within a larger circle. Rolling a circle of integral radius λ within a circle of integral radius κ,whereκ ą λ, yields the parametrization θ ÞÑpκ ´ λqeiθ ` λep1´κ{λqiθ of the hypocycloid centered at the origin, containing the point κ, and having precisely κ cusps.

Proof of Theorem 1.1. Computing the coefficients bk,j from (7) we find that bk,j “ δkj for k “ 0, 1,...,d´ 2, and bd´1,j “´1 for all j, from which (6) yields 1 gpz1,z2,...,zd´1q“z1 ` z2 ` ...` zd´1 ` . z1z2 ¨¨¨zd´1 The image of the function g : Td´1 Ñ C defined above is the filled hypocycloid corresponding to the parameters κ “ d and λ “ 1, as observed in [9, §3]. 

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1862 W. DUKE, S. R. GARCIA, AND B. LUTZ

Figure 9. Cyclic supercharacters σX of Z{pZ,whereX “ A1, whose graphs fill out |X|-hypocycloids.

Acknowledgments Special thanks to Trevor Hyde for many helpful comments.

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Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1555 E-mail address: [email protected] URL: http://www.math.ucla.edu/~wdduke/ Department of Mathematics, Pomona College, Claremont, California 91711 E-mail address: [email protected] URL: http://pages.pomona.edu/~sg064747 Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109-1043 E-mail address: [email protected] URL: http://www-personal.umich.edu/~boblutz

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