Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm
aa r X i v : . [ m a t h . N T ] N ov KLOOSTERMAN SUMS, ELLIPTIC CURVES, ANDIRREDUCIBLE POLYNOMIALS WITH PRESCRIBED TRACEAND NORM
MARKO MOISIO
Abstract.
Let F q ( q = p r ) be a finite field. In this paper the number of irre-ducible polynomials of degree m in F q [ x ] with prescribed trace and norm coeffi-cients is calculated in certain special cases and a general bound for that number isobtained improving the bound by Wan if m is small compared to q . As a corollary,sharp bounds are obtained for the number of elements in F q with prescribed traceand norm over F q improving the estimates by Katz in this special case. Moreover,a characterization of Kloosterman sums over F r divisible by three is given gener-alizing the earlier result by Charpin, Helleseth, and Zinoviev obtained only in thecase r odd. Finally, a new simple proof for the value distribution of a Kloostermansum over the field F r , first proved by Katz and Livne, is given. Introduction
Let F q be a finite field with q = p r , and let a, b ∈ F q , b = 0. Fairly little is knownabout the number P m ( a, b ) of irreducible polynomials p ( x ) = x m − ax m − + · · · +( − m b in F q [ x ]. Carlitz [1] obtained the asymptotic formula P m ( a, b ) = q m − mq ( q −
1) + O ( q m/ ) ( m → ∞ ) , and evaluated P b P m ( a, b ) where b runs over F ∗ q , and over the set of squares (resp.non-squares) in F ∗ q . Later Yucas [20] calculated elementarily the numbers P a P m ( a, b )and P b P m ( a, b ) where a, b run over F q .By the bijection p ( x ) ( − m b − x m p ( x ) we see that P m ( a, b ) equals the numberof irreducible monic polynomials of degree m in the arithmetic progression { cx + d + f ( x ) x | f ( x ) ∈ F q [ x ] } where c = ( − m +1 ab − and d = ( − m b − . Applying a general asymptotic boundon the number of primes on an arithmetic progression (see e.g. [16, p.40]) we actually Date : November 3, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Elliptic curve; Exponential sum; Finite Field; Irreducible polynomial;Kloosterman sum; Kronecker class number. have the asymptotic bound P m ( a, b ) = q m − m ( q −
1) + O (cid:16) q m/ m (cid:17) ( m → ∞ ) . Finally, Wan [18, Thm. 5.1] obtained the following effective bound:(1.1) (cid:12)(cid:12)(cid:12)(cid:12) P m ( a, b ) − q m − m ( q − (cid:12)(cid:12)(cid:12)(cid:12) ≤ m q m . For a more complete survey the reader is referred to [4].The bounds above are obtained by using Dirichlet L-series over F q [ x ] and theRiemann’s hypothesis for function fields over a finite field. In this paper we express P m ( a, b ) in terms of the numbers N t ( a, b ) of elements x ∈ F q t (with t | m ) satisfying T race ( x ) = a and N orm ( x ) = b ( T race, N orm are from F q m onto F q ), which, inturn, are expressed in terms of exponential sums. This opens up a possibility tocalculate P m ( a, b ) explicitly in certain special cases. Moreover, we shall obtain animprovement of the bound (1.1) if m is small compared to q , more precisely, if m ≤ ( q − a = 0 the bound is obtained elementarily, but if a = 0 this is doneby linking the problem to the number of solutions of certain system of equations,and making use of the Katz bound [11]:(1.2) (cid:12)(cid:12)(cid:12)(cid:12) N m ( a, b ) − q m − q ( q − (cid:12)(cid:12)(cid:12)(cid:12) ≤ mq m − , proved by using deep algebraic geometry.The Katz bound with m = 3 plays a significant role in the proof by Cohen andHuczynska [10] of the existence of a primitive free (normal) cubic polynomial with a ( = 0) and b fixed, which completed a general existence theorem (see also [3, 5, 4]).We shall improve the Katz bound in this case. In fact, we get sharp lower and upperbounds for N ( a, b ), and, as a corollary for P ( a, b ), by using only the Hasse-Weilbound for elliptic curves together with a simple divisibility argument.Another special case where the Katz bound can be improved is the case m = p k forsome k . Especially, if p = 3 (resp. p = 2) a result on the distribution of irreduciblecubic (resp. quartic) polynomials in F q [ x ] with trace and norm prescribed is obtainedin terms of Kronecker class numbers by using the known value distribution of aKloosterman sum over F q [12, 13].Next, necessary and sufficient conditions for a Kloosterman sum over F r divisibleby three is given. In the case r odd this result follows also from [6, Thm. 3]. Finally,a new proof for the value distribution of a Kloosterman sum over the field F r isgiven. The proof uses only elementary properties of elliptic curves together with aresult by Deuring [8] which lies deeper: the knowledge of the number of isomorphismclasses of elliptic curves over F q having q + 1 + t points with gcd( q, t ) = 1. Acknowledgment.
The author is indebted to the anonymous referee for helpfulcomments and suggestions which improved the clarity of the paper considerably.
RREDUCIBLE POLYNOMIALS AND KLOOSTERMAN SUMS 3 Basic formulae
The aim of this section is to establish a link between the numbers N m ( a, b ) and P m ( a, b ), and to give basic formulae for N m ( a, b ) and P m ( a, b ) in terms of exponentialsums. The formulae will be studied more closely in later sections. m, p, r fixed positive integers, m ≥ p a prime F q the finite field with p r elements a, b fixed elements in F q , b = 0 P m ( a, b ) the number of irreducible polynomials x m − ax m − + · · · + ( − m b ∈ F q [ x ] t a positive factor of md equals gcd( q − , mt ) γ t a primitive element of F q t g the primitive element of F q defined by g = Norm t ( γ t )tr t ( x ) the trace function from F q t onto F q Norm t ( x ) the norm function from F q t onto F q S t ( a, b ) the set of the elements x in F ∗ q t withtr m ( x ) = a and Norm m ( x ) = bN t ( a, b ) the number of elements in S t ( a, b ) µ the M¨obius function χ and e the canonical additive characters of F q and F q t X ( F q ) the set of rational points on an algebraiccurve X defined over F q The following two lemmas relate the numbers P m ( a, b ) and N t ( a, b ): Lemma 2.1. P m ( a, b ) = 1 m X t | m µ ( t ) N m/t ( a, b ) . Proof.
Let H t ( a, b ) = (cid:12)(cid:12) { x ∈ F ∗ q t | tr m ( x ) = a, Norm m ( x ) = b, and x F q s if s < t } (cid:12)(cid:12) . Obviously N m ( a, b ) = P t | m H t ( a, b ), and now by M¨obius inversion formula H m ( a, b ) = X t | m µ ( t ) N m/t ( a, b ) . But H m ( a, b ) = mP m ( a, b ) completing the proof. (cid:3) Lemma 2.2.
Let m = p e · · · p e k k be the canonical prime number decomposition of m ( p < p < . . . ), and let m ′ = p · · · p k . Then N m ( a, b ) − M m ′ / ≤ mP m ( a, b ) ≤ N m ( a, b ) + M ( m ′ − / MARKO MOISIO with M = max h { N m/h ( a, b ) } , M = max s { N m/s ( a, b ) } where h (resp. s > ) runsover the factors of m ′ having odd (resp. even) number of prime factors. If k = 1 ,set M = 0 .Proof. Assume k = 1. Now mP m ( a, b ) = N m ( a, b ) − N m/p ( a, b ) by Lemma 2.1.Moreover, since M = N m/p ( a, b ) and M = 0, the lemma follows in this case.Assume k >
1. By Lemma 2.1 we have mP m ( a, b ) = N m ( a, b ) + X s N m/s ( a, b ) − X h N m/h ( a, b ) . Since m ′ ≥ k , we now get mP m ( a, b ) − N m ( a, b ) ≥ − M X h − M ⌊ k/ ⌋ X i =0 (cid:18) k i + 1 (cid:19) = − M k − ≥ − M m ′ / . Moreover, mP m ( a, b ) − N m ( a, b ) ≤ M X s M ⌊ k/ ⌋ X i =1 (cid:18) k i (cid:19) = M (2 k − − ≤ M ( m ′ − / , and the proof is complete. (cid:3) Next we derive a formula for N t ( a, b ). First, we observe that if a = 0, then(2.1) x ∈ S t ( a, b ) ⇔ p ∤ mt and tr t ( x ) = tm a and Norm t ( x mt ) = b, and if a = 0, then x ∈ S t ( a, b ) ⇔ p | mt and Norm t ( x mt ) = b, (2.2) or, p ∤ mt , tr t ( x ) = 0 , and Norm t ( x mt ) = b. Second, we see thatNorm t ( x mt ) = b ⇔ mt i ≡ ind g b (mod q − ⇔ d | ind g b and i = i + q − d j where j runs over the set { , . . . , q t − q − /d − } , and i is a solution of the congruence mdt i ≡ ind g bd (mod q − d ). Lemma 2.3.
Assume p ∤ mt and d | ind g b . Let i be a solution of the congruence mdt i ≡ ind g bd (mod q − d ) and let a = tm a . Then N t ( a, b ) = dq ( q −
1) ( q t − σ t ( a, b )) , RREDUCIBLE POLYNOMIALS AND KLOOSTERMAN SUMS 5 where (2.4) σ t ( a, b ) = X c ∈ F ∗ q χ ( − ca ) X x ∈ F ∗ qt e ( cγ i t x q − d ) . Proof.
Let α be an element in F q t with tr t ( α ) = t/m . Now, by (2.3) and by theorthogonality of characters we get qN t ( a, b ) = qt − q − /d − X j =0 X c ∈ F q χ ( c tr m ( γ i + q − d jt − αa ))= X c ∈ F q χ ( − ca ) qt − q − /d − X j =0 e ( mt cγ i t γ q − d jt ) c tm c = dq − X c ∈ F q χ ( − ca ) X x ∈ F ∗ qt e ( cγ i t x q − d )= dq − q t − σ t ( a, b )) . (cid:3) Zero trace
In this section we assume that a = 0 and simplify formula (2.4) by using Gausssums and some very elementary group theory. This will enable us to obtain animprovement of the Katz bound and the Wan bound in the case a = 0. We use thefollowing notations: H n the subgroup of order n of the multiplicativecharacter group of F q λ the trivial character of H n For a multiplicative character ψ of F q t we define a Gauss sum G ( ψ ) := X x ∈ F ∗ qt e ( x ) ψ ( x ) . Lemma 3.1.
Let n be a factor of q − and let α ∈ F ∗ q t . Then X x ∈ F ∗ qt e ( αx n ) = X λ ∈ H n G (¯ λ ◦ Norm t ) λ (Norm t ( α )) , where ¯ λ = λ − . MARKO MOISIO
Proof.
It is easy to see [14, p.217] that X x ∈ F ∗ qt e ( αx n ) = X ψ ∈ H ′ n G ( ¯ ψ ) ψ ( α ) , where H ′ n is the subgroup of order n of the multiplicative character group of F q t .But the surjectivity of Norm t implies that H ′ n = { λ ◦ Norm t | λ ∈ H n } . (cid:3) Assume a = 0, p ∤ mt , and d | ind g b . Now, by Lemma 3.1 we get σ t (0 , b ) = X c ∈ F ∗ q X x ∈ F ∗ qt e ( cγ i t x q − d )= X c ∈ F ∗ q X λ ∈ H q − d G (¯ λ ◦ Norm t ) λ ( c t g i )= X λ ∈ H q − d G (¯ λ ◦ Norm t ) λ ( g i ) X c ∈ F ∗ q λ ( c n ) , where n = gcd( q − , t ). Since X c ∈ F ∗ q λ ( c n ) = X c ∈ F ∗ q λ n ( c ) = ( λ n = λ ,q − λ n = λ ⇔ if λ ∈ H n ∩ H q − d , we get σ t (0 , b ) = ( q − X λ ∈ H s G (¯ λ ◦ Norm t ) λ ( g i ) = ( q − X x ∈ F ∗ qt e ( γ i t x s ) , where s = gcd( n, q − d ). Thus, N t (0 , b ) = dq (cid:16) q t − q − X x ∈ F ∗ qt e ( γ i t x s ) (cid:17) , implying the following Theorem 3.2.
Assume p ∤ mt and d | ind g b . Then, N t (0 , b ) = d (cid:16) q t − − q − q X x ∈ F qt e ( γ i t x s ) (cid:17) , where s = gcd( t, q − d ) and d = gcd( mt , q − . Theorem 3.2 and the Weil bound (see e.g. [14, p.223]) imply an improvement ofthe Katz-bound (see (1.2)) in the case a = 0: Corollary 3.3. (cid:12)(cid:12)(cid:12)(cid:12) N m (0 , b ) − q m − − q − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( s − q m − , RREDUCIBLE POLYNOMIALS AND KLOOSTERMAN SUMS 7 where s = gcd( m, q − . We can now prove an improvement of the Wan bound (see (1.1)) in the case a = 0and m ≤ ( q − Corollary 3.4. (cid:12)(cid:12)(cid:12)(cid:12) P m (0 , b ) − q m − − m ( q − (cid:12)(cid:12)(cid:12)(cid:12) ≤ s − m q m − + q m − q − < q − q m , where s = gcd( m, q − .Proof. Since d ≤ m/t , it follows from (2.2) and (2.3) that the numbers M and M in Lemma 2.2 satisfy M < M ≤ p q m/p − q − ≤ q m/ − q − , and now, by Lemma 2.2 and Corollary 3.3, we get (cid:12)(cid:12)(cid:12)(cid:12) mP m (0 , b ) − q m − − q − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( s − q m − + m q m/ − q − . (cid:3) By Lemma 2.1, Theorem 3.2, (2.3), and (2.2) we get explicit expressions for P m (0 , b ) e.g. in the following special cases: Example 3.5.
If gcd( p, m, q −
1) = 1, then P m (0 , b ) = 1 m ( q − X t | m µ (cid:16) mt (cid:17) ( q t − − . Example 3.6. If m = p k >
1, then mP m (0 , b ) = q m − − q − − q m/p − q − . Non-zero trace
In this section we assume that a = 0. This case is much harder than the zerotrace case, and we are not able to find such a simple expression for N t ( a, b ) as in case a = 0. The best we can do is to give N t ( a, b ) in terms of the number of solutionsof a system of equations, and estimate that number by using the Katz-bound. Thismethod will lead us to get an improvement of the Wan bound also in the case a = 0. MARKO MOISIO
Let n = ( q − /d . By Lemma 3.1 and by substitution c
7→ − a − c we see that σ t ( a, b ) (see (2.4)) can be written in the form σ t ( a, b ) = X c ∈ F ∗ q χ ( c ) X λ ∈ H n G (¯ λ ◦ Norm) λ (cid:0) c t ( − a ) − t g i (cid:1) = X λ ∈ H n G (¯ λ ◦ Norm) λ (cid:0) g i ( − a ) − t (cid:1) X c ∈ F ∗ q χ ( c ) λ t ( c )= X λ ∈ H n G (¯ λ ◦ Norm) G ( λ t ) λ (cid:0) g i ( − a ) − t (cid:1) . Let c = g i a − t and use Davenport-Hasse Theorem [14, p.197] to get σ t ( a, b ) = ( − t − X λ ∈ H n G (¯ λ ) t G ( λ t ) λ (cid:0) ( − t c (cid:1) = ( − t − X λ ∈ H n G ( λ ) t G (¯ λ t )¯ λ (cid:0) ( − t c (cid:1) . Now, by the definition of a Gauss sum we get σ t ( a, b ) = ( − t − X x ,...,x t ,u ∈ F ∗ q χ ( x + · · · + x t + u ) X λ ∈ H n λ (cid:0) x · · · x t ( − u ) − t c − (cid:1) , and consequently, by substituting x
7→ − ux , . . . , x t
7→ − ux t , we obtain σ t ( a, b ) = ( − t − X x ,...,x t ,u ∈ F ∗ q χ ( − u ( x + · · · + x t − × X λ ∈ H n λ ( x · · · x t c − ) . We can now prove the following
Theorem 4.1. If a = 0 , p ∤ mt , and d | ind g b , then N t ( a, b ) = d ( q t − q ( q −
1) + ( − t − (cid:16) d − X i =0 N ( c i ) − d ( q − t q ( q − (cid:17) , where N ( c i ) is the number of solutions of ( x + · · · + x t = 1 x · · · x t = c i in F tq with c i = g q − d i + i a − t .Proof. Let n = ( q − /d and c = g i a − t . The orthogonality of characters impliesthat q ( q − N ( c i ) = X x ,...,x t ∈ F ∗ q X u ∈ F q χ ( u ( x + · · · + x t − X λ ∈ H q − λ ( c − i x · · · x t ) , RREDUCIBLE POLYNOMIALS AND KLOOSTERMAN SUMS 9 and consequently q ( q − d − X i =0 N ( c i ) = X x ,...,x t ∈ F ∗ q X u ∈ F q χ ( u ( x + · · · + x t − × X λ ∈ H q − d − X i =0 λ ( c − i x · · · x t ) . Here d − X i =0 λ ( c − i x · · · x t ) = λ ( c − x · · · x t ) d − X i =0 λ ( g − ni )= ( λ ( c − x · · · x t ) d if λ ∈ H n , , otherwise , and now, by (4.1), we get q ( q − d − X i =0 N ( c i ) = d X x ,...,x t ∈ F ∗ q X u ∈ F q χ ( u ( x + · · · + x t − × X λ ∈ H n λ ( c − x · · · x t )= d ( − t − σ t ( a, b ) + d X x ,...,x t ∈ F ∗ q X λ ∈ H n λ ( c − x · · · x t ) . Here X x ,...,x t ∈ F ∗ q X λ ∈ H n λ ( c − x · · · x t ) = X λ ∈ H n (cid:16) X x ∈ F ∗ q λ ( c − x ) (cid:17) t = ( q − t and it follows that σ t ( a, b ) = ( − t − (cid:16) q ( q − d d − X i =0 N ( c i ) − ( q − t (cid:17) . Lemma 2.3 now completes the proof. (cid:3)
Lemma 4.2.
Let n be a positive integer and let c ∈ F ∗ q . The number N ( c ) ofsolutions ( x , . . . , x n ) in F nq of ( x + · · · + x n = 1 x · · · x n = c satisfies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N ( c ) − ( q − n q ( q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ nq n − . Proof.
Choose m = t = n , and a = 1, b = c . Now d = gcd( mt , q −
1) = 1 and wechoose i = ind g b (see (2.3)). Now c = g i /a t , and by Theorem 4.1 we get N n ( a, b ) = q n − q ( q −
1) + N ( c ) − ( q − n q ( q − N ( c ) − ( q − n q ( q −
1) = N n ( a, b ) − q n − q ( q − . The Katz bound (1.2) now completes the proof. (cid:3)
We are now able to prove an improvement of the Wan bound (1.1) in the case a = 0 and m ≤ ( q − Corollary 4.3.
Let a, b ∈ F ∗ q . Then, (cid:12)(cid:12)(cid:12)(cid:12) P m ( a, b ) − q m − mq ( q − (cid:12)(cid:12)(cid:12)(cid:12) ≤ q m − + q m − q ( q −
1) + m q m − < q − q m . Proof. If p | mt or d ∤ ind g b , then N t ( a, b ) = 0 by (2.1) and (2.3). Assume p ∤ mt and d | ind g b . If t is even Theorem 4.1 implies N t ( a, b ) ≤ d ( q t − q ( q − − dN ( c ) + d ( q − t q ( q − c ∈ F ∗ q . Now, by Lemma 4.2 N t ( a, b ) ≤ d ( q t − q ( q − − d (cid:16) ( q − t q ( q − − tq t − (cid:17) + d ( q − t q ( q −
1) = d ( q t − q ( q −
1) + dtq t − . Since d ≤ m/t , we get(4.2) N t ( a, b ) ≤ m ( q t − tq ( q −
1) + mq t − . If t is odd, then N t ( a, b ) ≤ d ( q t − q ( q −
1) + dN ( c ) − d ( q − t q ( q − c ∈ F ∗ q , and N t ( a, b ) ≤ d ( q t − q ( q −
1) + d (cid:16) ( q − t q ( q −
1) + tq t − (cid:17) − d ( q − t q ( q −
1) = d ( q t − q ( q −
1) + dtq t − . Hence, bound (4.2) holds in this case too.Now, by (4.2), it is clear that the numbers M and M in Lemma 2.2 satisfy M < M ≤ q m − q ( q −
1) + mq m − , RREDUCIBLE POLYNOMIALS AND KLOOSTERMAN SUMS 11 and consequently, by Lemma 2.2 and by the Katz-bound (1.2), we get (cid:12)(cid:12)(cid:12)(cid:12) mP m ( a, b ) − q m − q ( q − (cid:12)(cid:12)(cid:12)(cid:12) < mq m − + m q m − q ( q −
1) + m q m − . Hence, (cid:12)(cid:12)(cid:12)(cid:12) P m ( a, b ) − q m − mq ( q − (cid:12)(cid:12)(cid:12)(cid:12) ≤ q m − + q m − q ( q −
1) + m q m − < q (cid:16) q − m q − m q (cid:17) q m = (cid:16) q − m q − m +44 (cid:17) q m . Obviously m q − m +44 < / ( q − (cid:3) Cubics and cubic extensions
In this section we assume that m = 3. Now the system of equations defined in theprevious section is of degree 3, and therefore we can give N ( a, b ), and also P ( a, b ),in terms of the number of rational points on a cubic curve defined over F q . Someelementary manipulations of cubic curves together with the Hasse-Weil bound forelliptic curves, and the link between P ( a, b ) and N ( a, b ), will then lead to a sharpbound for N ( a, b ), which is also an improvement of the Katz bound in the case m = 3. The following result is a key for such an improvement: Theorem 5.1.
Let c = ba − , and let X be the projective curve over F q defined by X : y + cy + xy = x . Then, N ( a, b ) = |X ( F q ) | and P ( a, b ) = ( |X ( F q ) | − ǫ ) , where ǫ = ( if p = 3 and c = , otherwise . Proof.
Let m = 3 and apply Theorem 4.1 with t = 3 to get(5.1) N ( a, b ) = q − q ( q −
1) + N ( c ) − ( q − q ( q −
1) = N ( c ) + 3 , where N ( c ) is the number of solutions ( x, y, z ) in F q of ( x + y + z = 1 xyz = c with c = g i /a = b/a , or, equivalently, N ( c ) is the number of solutions of(5.2) x y + xy − xy = − c , in F q .Equation (5.2) defines an affine component of the projective curve defined by X ′ : y + y − xy = − c x , and that affine component has exactly three points at infinity. Hence N ( a, b ) = |X ′ ( F q ) | by (5.1). By multiplying both sides of the equation of X ′ by c and then substituting x
7→ − c − x and y c − y we see that X ′ is isomorphic over F q to X .It follows from Lemma 2.1 that 3 P ( a, b ) = N ( a, b ) − N ( a, b ), and by (2.1) x ∈ S ( a, b ) if and only if p = 3 and b = ( a/ . This completes the proof. (cid:3) Corollary 5.2.
Assume p = 3 , and let b = ( a/ . Then P ( a, b ) = ( q ± , where the sign is plus if p ≡ and ∤ r , and otherwise the sign is minus.Proof. If p = 2 then the equation of X is y + ( x + 1) y = x . This equation hasonly three solutions with x = 0 ,
1. By substituting y ( x + 1) y , we can write theequation in the form y + y = x / ( x + 1) , and then by substitutintg x x + 1 weget the equation(5.3) y + y = x + 1 + x − + x − . Since the absolute trace of x − + x − equals zero we have χ ( x − + x − ) = 1, andtherefore equation (5.3) has exactly X x ∈ F ∗ q \{ } (1 + χ ( x + 1)) = q − χ (1) (cid:16) X x ∈ F ∗ q χ ( x ) − χ (1) (cid:17) = q − − χ (1)solutions in F q with x = 0 ,
1. Hence, in the case p = 2, |X ( F q ) | = q − − χ (1) + 3 + 1and P ( a, b ) = ( q − χ (1)) / p = 2 and write the equation y + cy + xy = x in the form ( y + ( c + x )) = x + ( c + x ) . Substitute y y − ( c + x ) to get y = x + x + cx + c = ( x + ) ( x + ) . Finally, by substituting x x − , we see that X is isomorphic over F q to C : y = x ( x − ) . Let F be the set of finite points of C and let F ′ be the set of finite points of thecurve C ′ defined over F q by C ′ : z = u − . We note that the map ( x, y ) ( u = x, z = yx ) from F \ { (0 , } to F ′ is injective,and it follows that | F | = | F | ′ ± z = − has,or has not, a solution in F q . Hence, | C ( F q ) | = | C ′ ( F q ) | ± q + 1 ±
1, and now, byTheorem 5.1, we get P ( a, b ) = ( q + 1 ± − (cid:3) RREDUCIBLE POLYNOMIALS AND KLOOSTERMAN SUMS 13
We can now prove an improvement of the Katz bound in the case m = 3: Theorem 5.3.
Let a, b ∈ F q , b = 0 . Then (cid:24) q + 1 − √ q (cid:25) ≤ N ( a, b ) ≤ (cid:22) q + 1 + 2 √ q (cid:23) . Proof.
By Lemma 2.1 we have3 P ( a, b ) = N ( a, b ) − N ( a, b ) . Assume first that a = 0. If p = 3 then N ( a, b ) = 1 by (2.2), and 3 P ( a, b ) = q +1 − N ( a, b ) = q + 1, and the theorem follows in the case a = 0and p = 3.If p = 3 then N ( a, b ) = 0 by (2.2), and Corollary 3.3 now implies(5.4) q + 1 − √ q ≤ P ( a, b ) ≤ q + 1 + 2 √ q, and therefore (cid:24) q + 1 − √ q (cid:25) ≤ P ( a, b ) ≤ (cid:22) q + 1 + 2 √ q (cid:23) . Since 3 P ( a, b ) = N ( a, b ), the proof is complete in case a = 0 and p = 3.Assume next that a = 0. It is easy to see that if X : y + cy + xy = x is singularthen p = 3 and c = . Hence, if X is singular then q ± P ( a, b ) = N ( a, b ) − N ( a, b ) = q ± X is singular.Assume that X is non-singular. Now, by the proof of Corollary 5.2, we see that p = 3 or c = , and therefore 3 P ( a, b ) = |X ( F q ) | = N ( a, b ) by Theorem 5.1. Now,since X elliptic, the Hasse-Weil bound (see e.g. [19, p.91]) implies that the boundsin (5.4) hold in this case too, and the proof is complete. (cid:3) Remark 5.4.
The bounds in Theorem 5.3 are sharp. Take q = 5, for example. If a = b = 1, we have N ( a, b ) = |X ( F q ) | = 9 = 3 ⌊ (5 + 1 + 2 √ / ⌋ . If a = 1, b = 2,we have N ( a, b ) = |X ( F q ) | = 3 = 3 ⌈ (5 + 1 − √ / ⌉ . These calculations can beverified e.g. by MAGMA.6. Degree a power of the characteristic
An improvement of the Katz bound can also be obtained in the special case m = p k ( >
2) as we shall see in this section. The key point is that in this case thenumber of solutions of our system of equations, and therefore N m ( a, b ) and P m ( a, b ),can be given in terms of hyper-Kloosterman sums over F q which can be estimatedby the Deligne bound obtained in [7] (see also [14, p.254]).In the special cases ( p, m ) = (3 , , (2 ,
4) we can go even further since then wecan use the known value distributions of Kloosterman sums to get fairly preciseinformation on the distribution of the irreducible cubic and quartic polynomials overthe fields F r and F r , respectively. These cases are condidered in subsections 6.1and 6.2. For a positive integer n and c in F ∗ q let k n ( c ) be an n -dimensional Kloostermansum (or a hyper-Kloosterman sum): k n ( c ) = X x ,...,x n ∈ F ∗ q χ (cid:18) x + · · · + x n + cx · · · x n (cid:19) . Theorem 6.1.
Assume m = p k > , and let a, b ∈ F ∗ q . Then, N m ( a, b ) = q m − − q − − m − k m − ( c ) , where c = b/a m . Moreover, (cid:12)(cid:12)(cid:12)(cid:12) N m ( a, b ) − q m − − q − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( m − q m − . Proof.
Apply Theorem 4.1 with m = t to get(6.1) N m ( a, b ) = q m − q ( q −
1) + ( − m − (cid:18) N ( c ) − ( q − m q ( q − (cid:19) , where N ( c ) is the number of solutions of ( x + · · · + x m = 1 x · · · x m = c Obviously N ( c ) is equal to the number of solutions of x + · · · + x m − + cx · · · x m − − , and therefore, by the orthogonality of characters, we get qN ( c ) = X x ,...,x m − ∈ F ∗ q X u ∈ F q χ (cid:18) u (cid:16) x + · · · + x m − + cx · · · x m − − (cid:17)(cid:19) = X u ∈ F ∗ q χ ( − u ) X x ,...,x m − ∈ F ∗ q χ (cid:18) ux + · · · + ux m − + ucx · · · x m − (cid:19) + ( q − m − . RREDUCIBLE POLYNOMIALS AND KLOOSTERMAN SUMS 15
Now, by substitutions x x /u, . . . , x m − x m − /u , and by noting that x x m is a permutation of F q , we get qN ( c ) − ( q − m − = X u ∈ F ∗ q χ ( − u ) X x ,...,x m − ∈ F ∗ q χ (cid:18) x + · · · + x m − + u m cx · · · x m − (cid:19) = X u ∈ F ∗ q χ ( − u ) X x ,...,x m − ∈ F ∗ q χ (cid:18) x m + · · · + x mm − + u m cx m · · · x mm − (cid:19) = X u ∈ F ∗ q χ ( − u ) X x ,...,x m − ∈ F ∗ q χ (cid:18)(cid:16) x + · · · + x m − + uc /m x · · · x m − (cid:17) m (cid:19) = X x ,...,x m − ∈ F ∗ q χ ( x + · · · + x m − ) X u ∈ F ∗ q χ (cid:18) u (cid:16) c /m x · · · x m − − (cid:17)(cid:19) . The inner sum equals q − − x · · · x m − is, or is not, equal to c /m . Hence, qN ( c ) − ( q − m − = qk m − ( c /m ) − X x ,...,x m − ∈ F ∗ q χ ( x + · · · + x m − )= qk m − ( c /m ) − ( − m − , and consequently N ( c ) = k m − ( c ) + 1 q (( q − m − − ( − m − ) , since k m − ( c /m ) = k m − ( c ). It now follows from (6.1) that N m ( a, b ) = q m − − q − − m − k m − ( c ) , and the Deligne bound concludes the proof. (cid:3) By Theorem 6.1, equation (2.1), and Lemma 2.1 we get an expression for P m ( a, b )in terms of a hyper-Kloosterman sum: Corollary 6.2. If m = p k > and ab = 0 , then mP m ( a, b ) = q m − − q − − m − k m − ( b/a m ) . Irreducible cubics over F r . Next we consider the number of irreduciblecubics P ( a, b ) when q = 3 r . The main result of this section is the following: Corollary 6.3.
Let q = 3 r and let a, b ∈ F q with ab = 0 . Then, P ( a, b ) = ( q + 1 + t ) / where t is an integer satisfying the following two conditions: (i) t ≡ − , (ii) | t | < √ q .Conversely, for a given integer t satisfying conditions (i) and (ii) there are exactly ( q − H ( t − q ) pairs ( a, b ) ∈ F q with ab = 0 and P ( a, b ) = ( q + 1 + t ) / . Here H ( d ) is the Kronecker class number of d .Proof. For a given c ∈ F ∗ q there are exactly q − a, b ) ∈ F q such that c = b/a .Corollary 6.2, Theorem 6.4, and Theorem 6.4 below complete the proof. (cid:3) Theorem 6.4 ([12]) . Let q = 3 r . The range S of k ( c ) , as c runs over F ∗ q , is givenby S = { t ∈ Z : | t | < √ q and t ≡ − } . Moreover, each value t ∈ S is attained exactly H ( t − q ) times. Example 6.5.
Let q = 3. If t is an integer satisfying conditions (i) and (ii) then t = − t = 2. There should be exactly (3 − H (1 −
12) = 2 pairs ( a, b ) with ab = 0 and P ( a, b ) = 1, and exactly (3 − H (4 −
12) = 2 pairs ( a, b ) with ab = 0and P ( a, b ) = 2.Indeed, the two pairs ( a, b ) for which there is exactly one irreducible polynomial x + ax + cx + b ∈ F [ x ] are ( a, b ) = (1 , , (2 , x + x + 2 x + 1 , x + 2 x + 2 x + 2 . The two pairs ( a, b ) for which there are exactly two irreducible cubics are ( a, b ) =(1 , , (2 ,
1) and the corresponding irreducible cubics are x + x + 2 , x + x + x + 2 , x + 2 x + 1 , x + 2 x + x + 1 . Finally, for a pair (0 , b ) there should be, by Example 3.6, exactly one irreduciblecubic. Indeed, the corresponding polynomials are x + 2 x + 1 , x + 2 x + 2 . Thus we have counted all the eight irreducible cubics in F [ x ].6.2. Irreducible quartics over F r . We conclude Section 6 by considering thenumber of irreducible quartics P ( a, b ) when q = 2 r . We need the following resultby Carlitz which links one- and two-dimensional Kloosterman sums: Theorem 6.6 ([2]) . Let c ∈ F ∗ q . Then, k ( c ) = k ( c ) − q. Now we are able to prove the main result of this section:
Corollary 6.7.
Let q = 2 r ( r > ) and let a, b ∈ F q with ab = 0 . Then, P ( a, b ) =( q + 2 q + 1 − t ) / where t is an integer satisfying the following two conditions: (i) t ≡ , (ii) 1 ≤ t < √ q .Conversely, for a given integer t satisfying conditions (i) and (ii) there are exactly ( q − H ( t − q ) pairs ( a, b ) ∈ F q with ab = 0 and P ( a, b ) = ( q + 2 q + 1 − t ) / . RREDUCIBLE POLYNOMIALS AND KLOOSTERMAN SUMS 17
Proof.
Let c = b/a . By Corollary 6.2 4 P ( a, b ) = q + q + 1 − k ( c ), and now, byTheorem 6.6, we get 4 P ( a, b ) = q + 2 q + 1 − k ( c ) . Theorem 6.8 below completes the proof. (cid:3)
Theorem 6.8 ([13]) . Let q = 2 r . The range S of k ( c ) , as c runs over F ∗ q , is givenby S = { t ∈ Z : | t | < √ q and t ≡ − } . Moreover, each value t ∈ S is attained exactly H ( t − q ) times. Example 6.9.
Let q = 4. Now t = 1 and t = 3 are the only integers satisfying (i)and (ii). There should be exactly (4 − H (1 −
16) = 6 pairs ( a, b ) with ab = 0 and P ( a, b ) = (16 + 8 + 1 − / − H (9 −
16) = 3 pairs ( a, b ) with ab = 0 and P ( a, b ) = (16 + 8 + 1 − / F = { , , α, β } , then the six pairs ( a, b ) for which there are exactly sixirreducible polynomials x + ax + · · · + b ∈ F [ x ] are( a, b ) = (1 , α ) , (1 , β ) , ( α, , ( α, β ) , ( β, , ( β, α ) , and the three pairs ( a, b ) for which there are exactly four irreducible quartics( a, b ) = (1 , , ( α, α ) , ( β, β ) . Finally, for a pair (0 , b ) there should be, by Example 3.6, exactly ( q + q + 1 − ( q + 1)) / · · · F [ x ].7. Divisibility modulo three of Kloosterman sums, q = 2 r Let q = 2 r . We consider the divisibility modulo three of Kloosterman sums k ( c ) := k ( c ). We use the following notations:Tr q s the trace function from F q onto F s A the set of elements a ∈ F q with Tr q ( a ) = 0 T ( b ) the number of irreducibles x + ax + cx + b ∈ F q [ x ]with b fixed and a runs over the set A We need the following
Theorem 7.1 ([15]) . Let α ∈ F ∗ q m . Then, X x ∈ F ∗ qm e ( αx q − ) = ( − m − ( q − k m − (Norm m ( α )) . Lemma 7.2.
Let b ∈ F ∗ q . Then, T ( b ) = (cid:16) ( q + 1 + k ( b ) ) − N ( b ) (cid:17) , where N ( b ) is equal to the number of solutions of x = b in A . Proof.
By Lemma 2.1(7.1) 3 T ( b ) = X a ∈ A N ( a, b ) − X a ∈ A N ( a, b ) . By (2.1) the latter sum is equal to N ( b ). Consider next the first sum. ApplyLemma 2.3 with t = m = 3 to get X a ∈ A N ( a, b ) = q − q −
1) + 1 q ( q − X a ∈ A σ ( a, b ) , where X a ∈ A σ ( a, b ) = X c ∈ F ∗ q X x ∈ F ∗ q e ( cγ i x q − ) X a ∈ A χ ( ca )= 12 X c ∈ F ∗ q X x ∈ F ∗ q e ( cγ i x q − ) X a ∈ F q χ ( c ( a + a )) . Since χ ( ca ) = χ ( c a ) the orthogonality of characters now implies X a ∈ A σ ( a, b ) = 12 X c ∈ F ∗ q X x ∈ F ∗ q e ( cγ i x q − ) X a ∈ F q χ (( c + c ) a )= q X x ∈ F ∗ q e ( γ i x q − ) . By Theorems 7.1 and 6.6 we now get X a ∈ A σ ( a, b ) = q ( q − k ( b ) = q ( q − k ( b ) − q )(note that Norm ( γ i ) = g i = b ), and therefore X a ∈ A N ( a, b ) = ( q + q + 1 + k ( b ) − q ) . Equation 7.1 now completes the proof. (cid:3)
Theorem 7.3.
Let q = 2 r , and let b ∈ F ∗ q . Then, 3 divides k ( b ) if and only if oneof the following condition holds (1) r is odd and Tr q ( √ b ) = 0(2) r is even, b = a for some a ∈ F q , and Tr q ( a ) = 0 Proof.
We have, by Lemma 7.2 ( q + 1 + k ( b ) ) − N ( b ) ≡ , or, equivalently, k ( b ) ≡ − N ( b ) − . RREDUCIBLE POLYNOMIALS AND KLOOSTERMAN SUMS 19
Hence, 3 | k ( b ) if and only if N ( b ) ≡ N ( b ) = 1. If r is odd,then x = b has unique solution x = √ b in F q and therefore N ( b ) = 1 if and only ifTr q ( √ b ) = 0.Assume r is even and let ζ ( ∈ F ) be a primitive third root of unity. Now N ( b ) = 1if and only if b = a and Tr q ( aζ i ) = 0, for some a ∈ F q and for unique i ∈ { , , } .It follows by the transitivity of Tr q that the latter condition is equivalent to Tr q ( a ) =0. (cid:3) Remark 7.4.
In the case r odd Theorem 7.3 follows also from [6, Thm. 3] provedby using different methods.8. A proof for the value distribution of a Kloosterman sum, q = 3 r The aim of this section is to give a fairly elementary proof for Theorem 6.4. Let q = 3 r , and let c ∈ F ∗ q . Let k ( c ) := k ( c ) and let X be the elliptic curve over F q defined by X : y + cy + xy = x . Lemma 8.1. |X ( F q ) | = q + 1 + k ( c ) and k ( c ) ≡ − . Proof.
Choose p = m = 3, and combine Theorem 5.1 and Corollary 6.2 to get |X ( F q ) | = 3 P (1 , c ) = q + 1 + k ( c ) . (cid:3) Lemma 8.2. X is isomorphic over F q to X ′ : y = x + x − c .Proof. Complete the square to get the equation of X in the form( y + x + c ) = x + ( x + c ) . Then, substitute y y − x − c , x x − c to get y = x + x − c , and then, substitute x x , y y to obtain( y − x − x − c ) = 0 . (cid:3) Let E be an elliptic curve over F q . Starting from the long Weierstrass form y + a xy + a y = x + a x + a x + a of the equation of E , it is easy to see (see e.g. [19, p.10]) that the equation of E canbe given in the form y = x + ax + cx + d. If a = 0 the substitution x x + e with e = c/a yields the equation y = x + ax + e + ae + ce + d, and therefore we may assume that the equation of E is one of the following(i) y = x + ax + b (ii) y = x + cx + b for some a, b, c ∈ F q . Since E is smooth we must have ab = 0 in case (i), and c = 0in case (ii).The j -invariant of E is given by j ( E ) = ( − a /b in case (i) , . Lemma 8.3.
Let |E ( F q ) | = q + 1 + t . The following three conditions are equivalent (1) E is supersingular, (2) j ( E ) = 0 , (3) 3 | t .Proof. See [19, p.75,p.121] (cid:3)
Assume next that E is ordinary (i.e. non-supersingular). We may now assumethat E is defined by E : y = x + ax + b. Lemma 8.4. If a is a square in F ∗ q then E is isomorphic over F q to X ′ : y = x + x + b/a , and |E ( F q ) | = q + 1 + t for some integer t with t ≡ − .If a is not a square, then |E ( F q ) | = 2( q + 1) − |X ′ ( F q ) | , and |E ( F q ) | = q + 1 + t for some integer t with t ≡ .Proof. If a = c for some c ∈ F ∗ q , the substitutions x ax , y c y yields theequation y = x + x + b/a . Assume next that a is not a square. Let η bethe quadratic character of F q with η (0) = 0. The number of solutions N of y = x + ax + b in ∈ F q is N = X x ∈ F q (1 + η ( x + ax + b )) = q + X x ∈ F q η ( x + ax + b ) . Now substitute x ax to obtain N = q + η ( a ) X x ∈ F q η ( x + x + b/a ) = q − X x ∈ F q η ( x + x + b/a ) , and consequently |E ( F q ) | = N + 1 = q + 1 − (cid:0) |X ′ ( F q ) | − ( q + 1) (cid:1) . The remaining assertions follow now immediately by Lemmas 8.2 and 8.1. (cid:3)
RREDUCIBLE POLYNOMIALS AND KLOOSTERMAN SUMS 21
Proof of Theorem 6.4 . Let t ≡ − − √ q, √ q ). By Theorem 8.5 below there exist exactly H ( t − q ) pairwise non-isomorphic elliptic curves E with |E ( F q ) | = q + 1 + t , and by Lemma 8.3 each of themis ordinary. Now, by Lemma 8.4 each E is isomorphic over F q to X ′ : y = x + x + c for some c ∈ F ∗ q , and finally Lemmas 8.2 and 8.1 conclude the proof. (cid:3) Theorem 8.5 ([8, 17]) . The number M ( t ) of isomorphism classes of elliptic curvesover F q having q + 1 + t points with gcd( q, t ) = 1 is given by M ( t ) = ( H ( t − q ) if t < q, otherwise . Remark 8.6.
Yet another proof of Theorem 6.4, which uses fairly advanced meth-ods, is given in [9].
References [1] L. Carlitz, A theorem of Dickson on irreducible polynomials, Proc. Amer. Math. Soc. 3 (1952)693-700.[2] L. Carlitz, A note on exponential sums, Pacific J. Math. 30 (1969) 35–37.[3] S.D. Cohen, Gauss sums and a sieve for generators of Galois fields, Publ. Math. Debrecen 56(2000) 293-312.[4] S.D. Cohen, Explicit theorems on generator polynomials, Finite Fields Appl. 11 (2005) 337–357.[5] S.D. Cohen, S. Huczynska, Primitive free quartics with specified norm and trace, Acta Arith.109 (2003) 359-385.[6] P. Charpin, T. Helleseth, V. Zinoviev, The divisibility modulo 24 of Kloosterman sums on GF (2 m ), m odd, J. Combin. Theory Ser. A 114 (2007) 322-338.[7] P. Deligne, Applications de la formule des traces aux sommes trigonom´etriques, in SGA 4 ,168–232, Lecture Notes in Math. 569, Springer-Verlag, Berlin (1977).[8] M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenk¨orper, Abh. Math.Sem. Hansischen Univ. 14 (1941) 197–272.[9] G. van der Geer, M. van der Vlugt, Artin-Schreier curves and codes, J. Algebra 139 (1991)256–272.[10] S. Huczynska, S.D. Cohen, Primitive free cubics with specified norm and trace, Trans. Amer.Math. Soc. 355 (2003) 3099-3116.[11] N.M. Katz, Estimates for Soto-Andrade sums, J. Reine Angew. Math. 438 (1993) 143-161.[12] N.M. Katz, R. Livne, Sommes de Kloosterman et courbes elliptiques universelles en caracter-istiques 2 et 3, C.R. Acad. Sci. Paris (I) 309 (1989) 723–726.[13] G. Lachaud, J. Wolfmann, The weights of orthogonals of the extended quadratic binary Goppacodes, IEEE Trans. Inform. Theory 36 (1990) 686–692.[14] R. Lidl, H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997.[15] M. Moisio, On the number of rational points on some families of Fermat curves over finitefields, Finite Fields Appl. 13 (2007) 546–562.[16] M. Rosen, Number theory in Function Fields, Springer, New York, 2002.[17] R. Schoof, Non-singular plane cubic curves over finite fields, J. Combin. Theory Ser. A 46(1987), 183–211.[18] D. Wan, Generators and irreducible polynomials over finite fields, Math. Comp. 219 (1997)1195-1212. [19] L.C. Washington, Elliptic Curves. Number Theory and Cryptography, Chapman andHall/CRC, Boca Raton, 2003.[20] J.L. Yucas, Irreducible polynomials over finite fields with prescribed trace/prescribed constantterm, Finite Fields Appl. 12 (2006) 211–221. E-mail address : [email protected]@uwasa.fi