Jacobi Sums Research Articles

On the number of solutions of some Kummer equations over finite fields

Let l be a prime number and let k=Fq be a finite field of characteristic p≠l with q=pf elements. Let n≥0. We determine the number N of solutions (x,y) in k of the Kummer equationyl=x(xln−1), in terms of the trace of a certain Jacobi sum.

ON THE p-ADIC VALUE OF JACOBI SUMS OVER F p 3

Let p be a prime and q = p s and ζk a fixed primitive kth root of unity in some extension of Q. Let χ be a multiplicative character over Fq of order k and J (χ, χ) the associated Jacobi sum. We give examples of χ which satisfy J (χ, χ) ∈p [s/2]Z[ζk]. Moreover, for s = 3, we prove that there is only a finite number of k such that J(χ, χ) is an element of pZ[ζk] except for the case where k is divisible by nine and p ≡1 ± k/3 (mod k).

Jacobi sum, differential equation modulo prime and logarithmic growth

We study the function Fd,m,ν=∫xν(1−xm)ddx for m∈N, d∈Q and 0⩽ν⩽m−1. We regard Fd,m,ν as a p-adic analytic function. We prove that Fd,m,ν satisfies a Frobenius equation and see that Jacobi sum appears in this equation. We consider the relation between the dimension dimkerbddp(L) of space of bounded solutions of L(y)=(D−νx+dmxm−11−xm)D(y)=0 and the dimension dimKerpL of the space of solutions of L in Fp[[x]]. We prove that dimkerbddp(L)⩽dimKerpL for almost all primes and that dimkerbddp(L)=2 for almost all primes if and only if dimKerpL=2 for almost all primes. Moreover in this case Fd,m,ν is an algebraic function.

Explicit Two-Sided Estimates for Norms of Strong Fejér–Jacobi Sums with Exponent at Most Two

We obtain two-sided estimates for the norms of strong Fejer–Jacobi sums with explicit constants. Bibliography: 9 titles.

The weight distributions of a class of cyclic codes III

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ma et al. (2011) [10], Ding et al. (2011) [6], Wang et al. (2012) [15], Xiong (2012) [16,17]. In this paper we solve one more special case. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which in turn can be solved by using the Jacobi sums directly.

The weight distributions of a class of cyclic codes II

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ding et al. (IEEE Trans Inform Theory 57(12), 8000–8006, 2011); Ma et al. (IEEE Trans Inform Theory 57(1):397–402, 2011); Wang et al. (Trans Inf Theory 58(12):7253–7259, 2012); and Xiong (Finite Fields Appl 18(5):933–945, 2012). In this paper we use the method developed in Xiong (Finite Fields Appl 18(5):933–945, 2012) to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which turns out to be associated with counting the number of points on some elliptic curves over finite fields. We also treat the special case that the characteristic of the finite field is 2.

利用Gauss 和与Jacobi 和构造近似MUB 和SIC-POVM

Mutually unbiased bases (MUB) and symmetric informationally complete positive operator-valued measure (SIC-POVM) are both important objects in quantum information theory. While people do not know if there exists a complete MUB for non-prime-power dimension, several versions of approximately MUB have been considered by relaxed the inner product condition. So far there are only finite number of <italic>K</italic> such that SIC- POVMs in Ck have been found. As in the MUB case, several versions of approximately SIC-POVM have been considered by relaxed the inner product condition. In this paper, we use the definitions of approximate MUB and SIC-POVM given by Klappenecker et al. For prime power q, we present simple constructions of <italic>q</italic> approximately MUB (AMUB) for dimension <italic>q</italic>-1, <italic>q</italic>+1 AMUB for dimension <italic>q</italic>-1, which shows the number of orthonormal bases of an AMUB in C^{<italic>k</italic>} can be more than <italic>K</italic>+1, and q AMUB for dimension <italic>q</italic>+1 by Gauss and Jacobi sums. We also present a construction of approximately SIC-POVM (ASIC-POVM) in dimension <italic>q</italic>-1 by Gauss sum.

Complementaries to Kummer's degree seven reciprocity law and a Dickson Diophantine system

Let ${\mathbb Q}(\zeta)$ be the cyclotomic field obtained from ${\mathbb Q}$ by adjoining a primitive seventh root of unity $\zeta$. Normalized primary elements of this field are characterized and related to Jacobi sums and to solutions of a system of quadratic Diophantine equations of Dickson type involving a rational prime $p\equiv 1\pmod{7}$. These objects and their connection are then used to give another formulation of the complementary laws to Kummer's reciprocity law of degree seven.

Two Classes of Codebooks Nearly Meeting the Welch Bound

In this paper, the value of Imax(C) for codebooks C = C(D) constructed by certain almost difference sets D in Fqx is determined and expressed in terms of Jacobi sums from which it shows that such codebooks nearly meet the Welch bound. This result is an answer of a question raised by C.Ding and T.Feng in [2]. We also present another series of codebooks which nearly meet the Welch bound, where the codebook is constructed by a subset R = Fq1 ⊕ Fq2. When q1 = q2, De is an almost difference set of R.

Jacobi Sum Matrices

In this article we identify several beautiful properties of Jacobi sums that become evident when these numbers are organized as a matrix and studied via the tools of linear algebra. In the process we reconsider a convention employed in computing Jacobi sum values by illustrating how these properties become less elegant or disappear entirely when the standard definition for Jacobi sums is utilized. We conclude with a conjecture regarding polynomials that factor in an unexpected manner.

EULER'S CRITERION FOR SEPTIC NONRESIDUES

Let p be a prime ≡ 1 (mod 7). In this paper, we obtain an explicit expression for a primitive seventh root of unity ( mod p) in terms of coefficients of a Jacobi sum of order 7 and also in terms of a solution of a Diophantine system of Leonard and Williams, and then obtain Euler's criterion for septic nonresidues D ( mod p) in terms of this seventh root. Explicit results are given for septic nonresidues for D = 2, 3, 5, 7.

Elementary computation of ramified components of the Jacobi sum Hecke characters

R. Coleman and W. McCallum calculated ramified components of the Jacobi sum Hecke characters explicitly using their computation of the stable reduction of the Fermat curve in Coleman and McCallum (1988) [CW]. In this paper, we give an elementary proof of the main result of them without using rigid geometry or the stable model of the Fermat curve.

Elliptic Gauss sums and applications to point counting

We define a class of algebras over finite fields, called polynomially cyclic algebras, which extend the class of abelian field extensions. We study the structure of these algebras; furthermore, we define and investigate properties of Lagrange resolvents and Gauss and Jacobi sums. Natural examples of polynomially cyclic algebras are for instance algebras of the form F p [ X ] / ( F q ( X ) ) where p , q are distinct odd primes and F q is the q - th cyclotomic polynomial. Further examples occur similarly on replacing the cyclotomic polynomials with factors of division polynomials of elliptic curves. Finally, Gauss and Jacobi sums over polynomially cyclic algebras are applied for improving current algorithms for counting the number of points of elliptic curves over finite fields.

On Jacobi sums in Q(ζp)

We study the arithmetic of Jacobi Sums Q(\zeta_p) and we show that they play a somehow parallel role in the minus part as to Units in the plus part of the Class Field Theory of Cyclotomic fields.

Hypergeometric $\mathbf {{}_3{\it F}_2(1/4)}$ evaluations over finite fields and Hecke eigenforms

Let H denote the hypergeometric 3 F 2 function over F p whose three numerator parameters are quadratic characters and whose two denominator parameters are trivial characters. In 1992, Koike posed the problem of evaluating H at the argument 1/4. This problem was solved by Ono in 1998. Ten years later, Evans and Greene extended Ono's result by evaluating an infinite family of 3 F 2 (1/4) over F q in terms of Jacobi sums. Here we present five new 3 F 2 (1/4) over F q (involving characters of orders 3, 4, 6, and 8) which are conjecturally evaluable in terms of eigenvalues for Hecke eigenforms of weights 2 and 3. There is ample numerical evidence for these evaluations. We motivate our conjectures by proving a connection between 3 F 2 (1/4) and twisted sums of traces of the third symmetric power of twisted Kloosterman sheaves.

On the Cross-Correlation Distributions of $M$-ary Multiplicative Character Sequences

It is well known that the magnitude of the cross correlation between any distinct constant multiple sequences of an M-ary power residue sequence of period p is upper bounded by radicp +2 and that of an M -ary Sidel'nikov sequence of period p m-1 is upper bounded by radic{p m} +3, where p is a prime and m is a positive integer. In this paper, we first show that their cross-correlation functions are closely related to Jacobi sums and cyclotomic numbers. We then derive the cross-correlation distribution of constant multiple sequences of an M -ary power residue sequence. In the case of constant multiple sequences of an M-ary Sidel'nikov sequence, we get the possible cross-correlation values whose occurrence numbers are expressed in terms of the cyclotomic numbers of order M and are possibly zero.

Jacobi and Kummer’s ideal numbers

In this article we give a modern interpretation of Kummer’s ideal numbers and show how they developed from Jacobi’s work on cyclotomy, in particular the methods for studying “Jacobi sums” which he presented in his lectures on number theory and cyclotomy in the winter semester 1836/37.

Perfect difference systems of sets and Jacobi sums

A perfect ( v , { k i ∣ 1 ≤ i ≤ s } , ρ ) difference system of sets (DSS) is a collection of s disjoint k i -subsets D i , 1 ≤ i ≤ s , of any finite abelian group G of order v such that every non-identity element of G appears exactly ρ times in the multiset { a − b ∣ a ∈ D i , b ∈ D j , 1 ≤ i ≠ j ≤ s } . In this paper, we give a necessary and sufficient condition in terms of Jacobi sums for a collection { D i ∣ 1 ≤ i ≤ s } defined in a finite field F q of order q = e f + 1 to be a perfect ( q , { k i ∣ 1 ≤ i ≤ s } , ρ ) -DSS, where each D i is a union of cyclotomic cosets of index e (and the zero 0 ∈ F q ). Also, we give numerical results for the cases e = 2 , 3 , and 4.

Pseudopowers and primality proving

It has been known since the 1930s that so-called pseudosquares yield a very powerful machinery for the primality testing of large integers N. In fact, assuming reasonable heuristics (which have been confirmed for numbers to 2^80) this gives a deterministic primality test in time O((lg N)^(3+o(1))), which many believe to be best possible. In the 1980s D.H. Lehmer posed a question tantamount to whether this could be extended to pseudo r-th powers. Very recently, this was accomplished for r=3. In fact, the results obtained indicate that r=3 might lead to an even more powerful algorithm than r=2. This naturally leads to the challenge if and how anything can be achieved for r>3. The extension from r = 2 to r = 3 relied on properties of the arithmetic of the Eisenstein ring of integers Z[\zeta_3], including the Law of Cubic Reciprocity. In this paper we present a generalization of our result for any odd prime r. The generalization is obtained by studying the properties of Gaussian and Jacobi sums in cyclotomic ring of integers, which are tools from which the r-th power Eisenstein Reciprocity Law is derived, rather than from the law itself. While r=3 seems to lead to a more efficient algorithm than r=2, we show that extending to any r>3 does not appear to lead to any further improvements.

Jacobi sums, Fermat Jacobians, and ranks of abelian varieties over towers of function fields

Our goal in this note is to give a number of examples of abelian varieties over function fields k(t) which have bounded ranks in towers of extensions such as k(t^{1/d}) for varying d. Along the way we prove some new results on Fermat curves which may be of independent interest.