1 THE QUINTIC GAUSS SUMS Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaflichen Fakult¨aten der Georg-August Universit¨at zu Goettingen vorgelegt von L´eopold Fossi Talom Aus Bandjoun/ Yom V, Kamerun G¨ottingen 2002 D7 Referent: Prof. S.J. Patterson Korreferentin: Frau Prof. Dr. I. Kersten. Tag der Disputation: Contents List of Figures 7 Chapter 1. INTRODUCTION 9 Chapter 2. Generalities on Cyclotomic Fields 17 1. Introduction 17 2. Power residue Symbol 24 3. The primary choice of the prime factor in Q(ζ5) 26 Chapter 3. GAUSS SUMS 29 1. Gauss sums in number fields 29 2. Factorization of Gauss sums 36 3. Gauss sums over finite fields and the Davenport-Hasse relations 39 4. The Gauss sum and Jacobi sum as complex numbers 40 5. The moment Gauss sums and the uniform distribution 45 Chapter 4. CYCLOTOMIC CRYSTAL 49 1. Introduction 49 2. The cyclotomic case 51 3. The vertices 56 4. The geometry and the combinatoric of the fundamental domain for n = 3,5 59 5. Generalization 65 6. Cyclotomy revisited 69 Chapter 5. THE GAUSS SUMS AND THE FUNDAMENTAL DOMAIN 75 1. First identity 77 2. Second identity 79 3. Third identity 80 4. Cassels' formula 82 5. A generalized of a formula of Cassels' type 83 6. Conjectures on the partial sums of the Gauss sums 89 Chapter 6. APPENDIX 105 1. The Gauss periods or the Lagrange resolvents 105 2. The quadratic Gauss sums 109 3. Algorithm for the computation of the Gauss sums 111 4. data 115 Bibliography 143 5 List of Figures 1 Fundamental domain for n=3 and 3-faces for n=5 74 1 Graphic of the arguments of Gauss sums for π 1 (mod λ3) N(π) 20000 ≡ 84 ≤ 2 Graphic of the arguments of Gauss sums for π 1 (mod λ3) N(π) 40000 ≡ 85 ≤ 3 frequency in each 1 degree on the circle 86 4 frequency in each 1 degree on the circle 87 5 both graphs plotted one against another 87 6 g(X) = n≤X Ω(S0; "; π) log(N(π)) 93 0<X<40:000 1 7 g(X) = P6=5 n≤X Ω(S0; "; π) log(N(π)) 93 X 0<X<40:000 1 8 g(X) = P n≤X Ω(S0; "; π) log(N(π)) 94 X 0<X<40:000 1 N(π) 9 g(X) = X P n≤X Ω(S0; "; π) log(N(π))(1 X ) 94 0<X<40:000 − 1 10 g(X) = P n≤X Ω(S0; "; π) log(N(π)) and f(X) = X 0<X<40:000 1 N(π) X n≤X P Ω(S0; "; π) log(N(π))(1 X ) 95 0<X<40:000 − P 1 N(π) 11 g(X) = X n≤X Ω(S0; "; π) log(N(π)) 1 X 0<X<40:000 − N(π) and n≤PX Ω(S0; "; π) log(N(π)) 1 X 95 0<X<40:000 − 1=2 1=2 12 f(X)P= X− n≤X; µ(n)n 97 0<X<50:000 1 1=2 13 f(X) = Pn≤X; µ(n)n 97 X 0<X<50:000 14 π≡1 (mod Pλ3); g(1; "; π) log(N(π)) N(π)≤X;X≤40000 N(π) Pand π≡1 (mod λ3); g(1; "; π) log(N(π))(1 X ) 99 N(π)≤X;X≤40000 − 1 P 3 15 X6=5 π≡1 (mod λ ); g(1; "; π) log(N(π)) N(π)≤X;X≤40000 6 N(π) 5 andPX− π≡1 (mod λ3); g(1; "; π) log(N(π))(1 X ) 100 N(π)≤X;X≤40000 − P g(1,ε,π) log(N(π)) 16 π≡1 (mod λ3); N(π) 100 N(π)≤X;X≤40000 17 P 6 g(1; "; π) log(N(π)) f(X) = X− 5 Ω(S0; "; π) π≡1 (mod λ3); N(π)≤X;X≤40000 7 8 LIST OF FIGURES 6 g(1; "; π) log(N(π)) N(π) and X− 5 1 Ω(S0; "; π) − X π≡1 (mod λ3); N(π)≤X;X≤40000 101 g(1,ε,π) log(N(π)) 18 π≡1 (mod λ3); 102 Ω(S0,ε,π)N(π) N(π)≤X;X≤40000 19 P 6 g(1; "; π) log(N(π)) X− 5 and N(π) π≡1 (mod λ3); N(π)≤X;X≤40000 102 CHAPTER 1 INTRODUCTION In 1801, Gauss in his Disquisitiones Arithmeticae introduced the following sum k 1 − 2πimn2=k G2(m; k) = e : n=0 X where m and k are coprime. This sum has a remarkably simple dependence on its arguments. In the case k odd, it has values as pk; or ipk: Gauss discovered in 1801 that in fact for k odd: pk if k 1 mod 4 G2(1; k) = ≡ ipk if k 1 mod 4 ≡ − He only found a proof 4 years later, in August 1805. He also determined these values for k 0; 2 mod 4: There are nowadays many different proofs of this theorem. See for example≡ [S.I], [K.L], [E.T] One important property is that Gauss sums generate quadratic subfields of cyclo- tomic fields. Indeed one can verify easily that for integer k odd, 2 k−1 G (m; k) = ( 1) 2 k = k∗: 2 − k−1 2πi Therefore the quadratic field Q( (( 1) 2 k)) is contained in Q(e k ): This fact is − one of the properties discovered bqy Gauss before he determined the correct sign for the Gauss sums. He used it to prove the law of quadratic reciprocity. He considers : for two distinct primes p; q, and if ( : ) is the Legendre symbol, then the expression q 1 2 q−1 q−1 p∗ G (1; p) − G(q; p) 2 = p∗ 2 mod q 2 ≡ ≡ q thus (0.1) G (1; p)q G(q; p) (mod p) 2 ≡ q (0.2) G(1; p) (mod p) ≡ p p (0.3) ∗ G(1; p) (mod p) ≡ q q and so multiply both side of the last equation above by p we obtain the desired result: q p q−1 p−1 = ( 1)( 2 )( 2 ): p q − Another proof is obtained as follows. The Chinese Remainder Theorem give the values of Gauss sums for composite odd numbers pq and the definition of G2(1; p) 9 10 1. INTRODUCTION for different values of prime p is known for congruence classes modulo 4 of the odd numbers p, q, and pq so that in plugging in all that within the following identity: p q G (1; pq) = ( )( )G (1; p)G (1; q); 2 q p 2 2 we deduce the law of quadratic reciprocity as: p q p−1 q−1 ( )( ) = ( 1) 2 2 : q p − This is another proof of the law of quadratic reciprocity. A more general Gauss sum is the following. k 1 − (2πim`n=k) Gn(m; k) = e : X`=0 We see that for k prime and gcd(k; m) = 1 Gn(m; k) = τn(χ); χ:χ=1 X6 where the χ are all non trivial character modulo k of order ` and τn(χ) = χ(a) exp(2πian=k) a Xmod k The cubic Gauss sum, also called Kummer sum after E.E. Kummer, is p 1 − 2πin3 G (1; p) = exp( ) 3 p n=1 X can be expressed as G3 = τ3(χ) + τ3(χ) where χ is a cubic character on Fp, and p 1 − 2πin τ (χ) = χ(n) exp( ): 3 p n=1 X 3 The sum G3(1; p)is indeed a root of the cubic period polynomial fp(X) = X 3pX pL where 4p = L2 + 27M 2. This polynomial has exactly one root in eac−h − of the interval [ 2pp; pp], [ pp; pp] and [pp; 2pp], and the question here is to − − − know which of them contains the root G3. The quotient τ3(χ) τ (χ) j 3 j lies on the unit circle. Looking at their arguments θp, Kummer observed here that the investigation on the 45 cases of prime p less than 500 gives the frequency 3:2:1 respectively and he suggested that Whx x χh( θp ) = + o( ) h = 1; 2; 3; p≡1 mod 3; j j 2 log x log x Xp<x π where χ1 (resp. χ2; χ3) is the characteristic function of the interval (0; 3 ] (resp. π 2π π 1 1 1 ( 3 ; 3 ]; ( 3 ; π] and Wh = 2 ; 3 ; 6 for h = 1; 2; 3 respectively. The study of the frequence 1. INTRODUCTION 11 analysis of this arguments was done by Lehmer [EL], Goldstine and Von Neumann [G.VN]. They found, for p < 1000 a different distribution approximately 4 : 3 : 2. This led one to doubt about the conjecture of Kummer. The same investigation led one observed that the arguments θp are likely equidistributed in the interval (0; π) for the Lebesgue measure. Then following a suggestion of Davenport [Da], Moreno [MO] in 1974 proved that if I (0; π] is a subinterval, then 2 I x x χ (3θ ) = j j + o( ) I p 2 log x log x p 1 mod 3;p<x ≡ X where I is the Lebesgue measure of I. This just proves that the arguments of j j3 the τ3(χ) are uniformly distributed in the three intervals listed above. The idea of Moreno is that, the character χ = π is associated to a factor π p, such that : j π 1 mod 3 in the field of cubic roots of unity Q(ζ3).