NOTES ON CHARACTER SUMS

WEN WANG

Abstract. In this article, the properties of character sums, including Gauss sums and Jacobi sums are investigated.

1. Introduction–Historical Notes

The origin of the Gauss sum and Jacobi sum in the work of C.F. Gauss and C.G.J. Jacobi. Gauss introduced the Gauss sum in his Disquisitione Arithmeticae[Ga1] in July, 1801, and Jacobi introduced the Jacobi sum in a letter to Gauss[Ja1] dated February 8, 1827.

The sum introduced by Gauss in 1801 is 푘∑−1 푒2휋푖푚푛2/푘, 푛=0 which is now called a quadratic Gauss sum. This sum is not easy to evaluate, even in the special case that 푚 = 1 and 푘 is an odd positive integer.√ In√ this case, Gauss was easily able to show that this sum has the value ± 푘 or ±푖 푘, according as 푘 is of the form 4푢 + 1 or 4푢 + 3, respectively. Specific examples convinced Gauss that the plus sign is always correct. On August 30, 1805, Gauss wrote in his diary he was able to prove his conjecture on the sign of these sums. A few years later, Gauss[Ga2] published an evaluation of his quadratic Gauss sum for all positive integer 푘.

In his study of primes in arithmetic progressions, G.L. Dirichlet [Di1] introduced the multiplicative character 휒 modulo 푘 and the sum

푘∑−1 퐺(휒) = 푒2휋푖푚푛/푘. 푛=0 This is also called a Gauss sum, as it coincides with the quadratic Gauss sum above in the case that 휒 has order 2 and 푘 is a prime not dividing 푚.

The sum now called a Jacobi sum, which is in essence the one Jacobi introduced in 1827, is

Date: 2010-03-12. Key words and phrases. Character sums, Finite fields. The calculation in section 9 of this article is done by SAGE, an open source mathematical calculation software program directed by William Stein. See http://www.sagemath.org/. 1 2 WEN WANG

∑ 퐽(휒, 휓) = 휒(푛)휓(1 − 푛), 푛 mod 푝 where 휒 and 휓 are multiplicative characters modulo a prime 푝. Jacobi was already aware in 1827 that Gauss and Jacobi sums are related like gamma and beta functions, i.e., for 푘 = 푝 and 푚 = 1.

퐽(휒, 휓) = 퐺(휒)퐺(휓)/퐺(휒휓) when 휒휓 is non-trivial.

After the initial work of Gauss, Dirichlet, and Jacobi, many well-known math- ematicians made contributions to the theory of Gauss and Jacobi sums, including L. Carlitz, A. Cauchy, S. Chowla, H. Davenport, G. Eisenstein, B. Gross, H. Hasse, N.M. Katz, N. Koblitz, L. Kronecker, E.E. Kummer, D.H. and E. Lehmer, I.J. Mordell, S.J. Patterson, C.L. Siegel, L. Stickelbegger, and A. Weil.

In the following sections, Gauss and Jacobi sums are introduced via the language of characters over finite fields.

2. Characters of finite fields

Suppose 퐹 be a finite field with 푞 = 푝푑 elements, where 푝 is a prime number. It is obvious that 픽푝 = ℤ/ℤ푝 and therefore 퐹 is a dimension 푑 vector space over ∗ 픽푝. We say that 휒 is a character of 퐹 , or just of 퐹 if there is no ambiguity, if 휒 is a multiplicative group homomorphism of 퐹 into ℂ∗. 휒 is called primitive if it is injective. Note that 퐹 ∗ is a cyclic group with a generater, say, 푔. Then 휒 is determined by 휒(푔) ∈ ℂ∗. If 휒 is primitive, one may choose 푔 such that 2휋푖 ∗ ∗ 휒(푔) = 푒 푝−1 . Note that Hom(퐹 , ℂ ) has a natural abelian group structure, i.e., for 휒, 휓 ∈ Hom(퐹 ∗, ℂ∗), 휒휓(푎) = 휒(푎)휓(푎), for any 푎 ∈ 퐹 ∗. It is not hard to verify that the identity for Hom(퐹 ∗, ℂ∗) is the character that maps every element of 퐹 ∗ to 1 ∈ ℂ, we shall call this character the identity character and denote it by 1.

For a given character, let 푚 be the smallest positive integer such that 휒푚 = 1. Hence 푚 divides 푞 − 1. It follows that there is a primitive order 푚 character of 퐹 if and only if 푚 divides 푞 − 1. For example, if 퐹 is a prime field and char퐹 > 2, then 2 ∣ 푝 − 1, therefore there exist an order 2 character of 퐹 . Since 퐹 ∗ is cyclic, therefore( this) order 2 character is unique, which is the well-known the Legendre ⋅ symbol: 푝 , which maps a quadratic residue to 1 and a non-residue to −1.

We say that 휏 is an additive character if 휏 is a group homomorphism from the additive group of 퐹 into the multiplicative group ℂ∗. Note that the trace map tr is an additive map, which will be used for the construction of the Gauss sum in the next section. MATH 414 COURSE PROJECT 3

3. Gauss Sums

Let 퐹 be a finite field with 푞 elements, and let tr be the trace map from 퐹 to the prime field 픽푝. The Gauss sum of a character 휒 of 퐹 is defined as follows: ∑ tr푡 퐺(휒) := 휒(푡)휁푝 . 푡∈퐹 In general, according to an additive character 휏, we can form the Gauss sum in the following way: ∑ 퐺푎(휒; 휏) = 휒(푡)휏(푎푡) 푡∈퐹 for some fixed 푎 ∕= 0 ∈ 퐹 .

Note that if we consider the case when 퐹 itself is the prime field 픽푝, then the Gauss sum becomes: ∑ 푎푡 퐺푎(휒) := 휒(푡)휁푝 . 푡∈퐹 For now on and to the end of this section we will consider the case when 퐹 is the prime field. Note that 휒(푡) is a power of 휁푝−1 for any 휒 and any 푡 ∈ 퐹 , therefore the Gauss sum 퐺(휒) ∈ ℚ(휁푝−1, 휁푝). We also have the following basic property of the Gauss sums[Ir1, Co1]:

√ Proposition 3.1. If 휒 ∕= 휀, then ∣퐺(휒)∣ = 푝. Proof. The ideal is to evaluate the sum ∑ 퐺푎(휒)퐺푎(휒) 푎 in two ways. −1 If 푎 ∕= 0, then 퐺푎(휒) = 휒(푎)퐺(휒) and 퐺푎(휒) = 휒(푎) 퐺(휒). Thus 퐺푎(휒)퐺푎(휒) = 2 2 ∣퐺(휒)∣ . Since 퐺0 = 0, the sum evaluates as (푝 − 1)∣퐺(휒)∣ . On the other hand, ∑ ∑ 푎푥−푎푦 퐺푎(휒)퐺푎(휒) = 휒(푥)휒(푦)휁 . 푥 푦 Summing both sides over 푎 yields, ∑ ∑ 푠푢푚푎퐺푎(휒)퐺푎(휒) = 휒(푥)휒(푦)훿(푥, 푦)푝 = (푝 − 1)푝, 푥 푦 where 훿(푥, 푦) = 1 if 푥 = 푦 and zero otherwise. √ Hence ∣퐺(휒)∣ = 푝. □ 4 WEN WANG

4. Jacobi Sums

The Jacobi sum was first introduced by Jacobi in a letter to Gauss, which is ∗ defined as following. For two multiplicative character on 픽푝, we defined the Jacobi sum to be ∑ 퐽(휒, 휓) = 푡 휒(푡)휓(1 − 푡) Here are some basic properties of the Jacobi sum:

Proposition 4.1. Let 휀 be the trivial character. Then, (a) 퐽(휀, 휀) = 푝. (b) 퐽(휀, 휒) = 0. (c) 퐽(휒, 휒−1) = −휒(−1). (d) If 휒휓 ∕= 휀, then

퐺(휒)퐺(휓) 퐽(휒, 휓) = . 퐺(휒휓) Proof. Part (a) is immediate. Part (b) is a consequence of Lemma 7.3.

To prove (c), notice that ( ) ∑ ∑ (푎) ∑ 푎 퐽(휒, 휒−1) = 휒(푎)휒−1(푏) = 휒 = 휒 . 푏 1 − 푎 푎+푏=1 푎+푏=1,푏∕=0 푎∕=1 Set 푎/(1 − 푎) = 푐. If 푐 ∕= 1, then 푎 = 푐/(1 + 푐). It follows that as 푎 varies over 픽푝, except the element 1, then 푐 varies over 픽푝, except for −1. Thus ∑ 퐽(휒, 휒−1) = 휒(푐) = −휒(−1). 푐∕=−1 To prove (d), notice that ( )( ) ∑ ∑ 퐺(휒)퐺(휓) = 휒(푥)휁푥 휓(푦)휁(푦) 푥 푦 ∑ = 휒(푥)휓(푦)휁푥+푦 푥,푦 ( ) ∑ ∑ = 휒(푥)휓(푦) 휁푡. 푡 푥+푦=푡

If 푡 = 0, then ∑ ∑ ∑ 휒(푥)휓(푦) = 휒(푥)휓(−푥) = 휓(−1) 휒휓(푥) = 0, 푥+푦=0 푥 푥 since 휒휓 ∕= 휀 by assumption.

If 푡 ∕= 0, define 푥′ and 푦′ by 푥 = 푡푥′ and 푦 = 푡푦′. If 푥 + 푦 = 푡, then 푥′ + 푦′ = 1. It follows that MATH 414 COURSE PROJECT 5

∑ 퐺(휒)퐺(휓) = 휒휓(푡)퐽(휒, 휓)휁푡 = 퐽(휒, 휓)퐺(휒휓). 푡 Substituting into Equation (7.1) yields

퐺(휒)퐺(휓) = 퐽(휒, 휓)퐺(휒휓). □

We have the following consequence

√ Corollary 4.2. If 휒, 휓, 휒휓 are all not equal to 휀, then ∣퐽(휒, 휓)∣ = 푝.

Proof. Clear from Proposition 4.1. □

5. The Analogy between Gauss and Jacobi sums and Gamma and Beta functions

Gross and Koblitz [Ko1, Ko2]noted that there is an analogy between the Gauss sums and Jacobi sums, to the Gamma and Beta functions respectively. In this section the analogy is stated briefly.

Recall the (real) gamma function on 푥 > 0 is defined as:

∫ ∞ Γ(푥) = 푡푥−1푒−푡푑푡, 0 and the (real) beta function on 푥 > 0, 푦 > 0 is defined as:

∫ 1 퐵(푥, 푦) = 푡푥−1(1 − 푡)푥−1푑푡. 0

Note that the relation between Beta and Gamma function could be written as:

Γ(푥)Γ(푦) 퐵(푥, 푦) = Γ(푥 + 푦) ∫ Here comes the analogy: the integration 푑푡 could be viewed as a sum on the domain (0, 1). Note that 푡 7→ 푡푥−1 is a multiplicative character from the multi- −푡 plicative group of ℝ≥0 to itself, determined by 푥. Also note that 푒 is an additive character from [0, +∞) to [0, 1]. The analogy is then visually clear. 6 WEN WANG

6. Eisenstein Sums

Let 픽푝 = 푘 ⊂ 퐾 ⊂ 퐹 be a tower of extensions of finite fields. Let 퐾 = 픽푞 and 푓 퐹 = 픽푞푛 and put 푞 = 푝 . Then 퐹 is a finite extension of 퐾 and 푘. Denote with Tr the trace from 퐹 to 푘, and tr the trace form 퐾 to 푘, and T the trace from 퐹 to 퐾. Note that Tr = T ∘ tr.

For a character 휒 on 퐹 with order 푚 > 1 and an element 훼 ∈ 퐾, we could form the Eisenstein sum ∑ 퐸(휒; 훼) = 휒(푡). 푇 (푡)=훼

It is clear that 퐸(휒; 훼) ∈ ℤ[휁푚]. If 훼 = 푎 ∈ 퐹 , we put 퐸(휒; 훼) = 퐸푎(휒), ∗ 퐸(휒) = 퐸1(휒). Denote by 휒퐾 be the restriction of 휒 on 퐾 . The following properties of Eisenstein sums follows immediately from definition[?]:

Proposition 6.1. Let 휒 be a character defined on 퐹 ∗, where 퐹 is the extension of degree 푛 over 퐾 = 픽푞. (1) 퐸(휒; 훼) = 휒(훼)퐸(휒) for all 훼 ∈ 퐾∗; (2) 퐸(휒푝; 훼) = 퐸(휒; 훼푝); (3) 퐺훼(휒) = 퐸(휒)퐺훼(휒퐾 ) + 퐸0(휒) for all 훼 ∈ 퐾; (4) The table below gives some useful relations between Gauss and Eisenstein sums depending on whether 휒퐾 = 1퐾 or not:

−1 퐸0(휒) 퐺훼(휒) 퐸(휒)퐸(휒 ) 푛−1 휒퐾 ∕= 1퐾 0 퐸(휒)퐺훼(휒) 푞 푛−2 휒퐾 = 1퐾 (1 − 푞)퐸(휒) 푞퐸(휒) 푞

7. Power Residue Characters of Small Primes

Recall that the definition of Jacobi sums gives rise to the following proposition[Le1]:

Proposition 7.1. Let 푝 = 푚푛 + 1 be a prime, and let 휒 be a character of order ∗ 푛 푛 푚 on 픽푝. Then 휒(2) = −퐽(휒 , 휒 ) mod 2. In particular, if 푚 is odd then 2 is an 푚-th power residue modulo 푝 if and only if 퐽(휒, 휒) ≡ 1 mod 2. Proof. The key observation is that the summation over 푡 in the Jacobi sum ∑푝−2 −퐽(휒푛, 휒푛) = 휒푛(푡)휒푛(1 − 푡) 푡=2 푝+1 is symmetric with respect to 2 : since the contributions from 푡 and 푝 + 푡 − 1 are equal and therefore cancel modulo 2, we find

(푝 + 1) ( 푝 + 1) −퐽(휒푛, 휒푛) ≡ 휒푛 휒푛 1 − = 휒−2푛(2) = 휒(2) mod 2. 2 2 MATH 414 COURSE PROJECT 7

Since different 푚-th roots of unity differ modulo 2 when 푚 is odd, this implies 푛 푛 푛 푛 that 휒(2) = 1 if and only if 퐽(휒 , 휒 ) ≡ 1 mod 2. But 퐽(휒 , 휒 ) = 휎푛퐽(휒, 휒), 푛 where 휎푛 is the automorphism of ℚ(휁푚) that sends 휁푚 7→ 휁푚, and our claim follows. □

We also have the following Proposition that characterizes the power residue modulo small primes ([Le1], Prop.4.28.).

Proposition 7.2. Let 픭 be a prime ideal( ) of degree 1 in 퐾 = ℚ(휁푚); its norm is ⋅ then a prime 푝 ≡ 1 mod 푚. Let 휒 = 푝 be an 푚-th power character modulo 픭. 푚 Then 퐽(휒푎, 휒푏) ≡ 0 mod 픭 for all integers 푎, 푏 ≥ 1 such that 푎 + 푏 ≤ 푚 − 1. Proof. We have 휒(푡) ≡ 푡(푝−1)/푚 mod 픭 for every integer 푡 coprime to 푝 by definition of the power residue symbol. Hence

퐽(휒푎, 휒푏) = 푡푎(푝−1)/푚(1 − 푡)푏(푝−1)/푚 mod 푝. The lemma below shows that the right hand side is divisible by 푝 whenever the degree (푎 + 푏)(푝 − 1)/푚 of the polynomial is strictly smaller than 푝 − 1. Since 픭 ∣ 푝, our claim follows. □

Lemma 7.3. Let 푝 be a prime; then

푝−1 { ∑ 0 mod 푝, if 0 < 푘 < 푝 − 1; 푎푘 ≡ −1 mod 푝, if 푘 = 푝 − 1. 푎=1 Proof. □

Lemma 7.4. Let 휒 be a character on 픽푞, and let 휌 be the quadratic character on 픽푞, then 퐽(휒, 휌) = 휒(4)퐽(휒, 휒).

8. Number of Points on Certain Elliptic Curves and Jacobi Sums

A good use of the Jacobi sum is to determine the number of points on diagonal surfaces. As an example, here we evaluate the number of points of the special elliptic curves

퐸 : 푦2 = 푥3 + 퐷

Let 휌 be the character of 픽푝 of order 2 and 휒 be the character of 픽푝 with order 3. Then the number of points on 퐸1 could be written as 8 WEN WANG

∑ 푁(푦2 = 푥3 + 퐷) = 푁(푦2 = 푢)푁(푥3 = −푣) 푢+푣=퐷 ∑ = (1 + 휌(푢))(1 + 휒(−푣) + 휒(−푣)2) 푢+푣=퐷 ∑ ∑ = 푝 + 휌(푢)휒(푣) + 휌(푢)휒2(푣) 푢+푣=퐷 푢+푣=퐷 = 푝 + 1 + 휌휒(퐷) + 휌휒(퐷)퐽(휌, 휒)

By applying the fact 퐽(휌, 휒) = 휒(4)퐽(휒, 휒), one gets

푁(푦2 = 푥3 + 퐷) = 푝 + 1 + 휌휒(4퐷)퐽(휒, 휒) + 휌휒(4퐷)퐽(휒, 휒).

2 3 For example let us consider the curve 푦 = 푥 + 5 over 픽19. Note that 19 = (5 + 3휔)(3 + 3휔2), where 휔 is a primitive third root of unity. Note that 5 + 3휔 ≡ 2 mod 3, then 퐽(휒, 휒) = 5 + 3휔. Since 4 ⋅ 5 = 1 is sixth power in 픽19, therefore the number of points is given by 19 + 1 + 5 + 3휔 + 5 + 3휔2 = 27. MATH 414 COURSE PROJECT 9

References ( ) Ch1. S. Chowla, B. Dwork, and R.J. Evans. On the mod 푝2 determination of (푝−1)/2 . J. Number (푝−1)/4 Theory24(1986), 188-196. Co1. H. Cohen, Number Theory: Volume I Tools and Diophantine Equations, 2ed. Graduate Texts in Mathematics 239. Springer-Verlag, Newyork, 1997. Di1. G.L. Dirichlet. Recherches sur diverses applications de l’analysis infinit´esimale`ala th´eorie des nobres, J. Reine Angew. Math.21(1840), 134-155. Ga1. C.F. Gauss, Disquisitiones Arithmeticae, Fleischer, Leipzig, 1801. Ga2. C.F. Gauss. Summatio quarumdam serieum singularium, Comment. Soc. Reg. Sci. Gottin- gensis1(1811), 18pp. Go1. B.H. Gross, N. Koblitz. Gauss sums and the 푝-adic Γ-function. Ann. Math. 109(1979), 569- 581. Ir1. K.F. Ireland, M.I. Rosen. A Classical Introduction to Modern Number Thoery, 2ed . Grad- uate Texts in Mathematics 84. Springer-Verlag, New York, 1990. Ja1. C.G.J. Jacobi, Brief an Gauss vom 8. Februar 1827. [CW: vol. 7, pp.393-400] Ko1. N. Koblitz. p-adic Numbers, p-adic Analysis, and zeta-Functions. Graduate Texts in Math- ematics 59. Springer-Verlag, New York, 1977. Ko2. N. Koblitz. p-adic Analysis, a short Course on Recent Work. London Mathematical Society lecture notes series 46. Cambirge University Press, London-New York, 1980. Le1. F. Lemmermeyer. Reciprocity laws: from Euler to Eisenstein.Springer-Verlag, Berlin- Heidelberg-New York, 2000. Sc1. C.-G. Schmidt. Gause Sums and the Classical Gamma Function. Bull. London Math. Soc.12 (1980), 344-346.

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