Jacobi Sums Research Articles

The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4

In a cyclotomic scheme over a finite field, there are some relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the field. These relations were investigated in [3]. In this paper, we replace the finite field by a commutative local ring which is called a Galois ring of characteristic 4. Hence we want to find similar relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the local ring. Specifically, if we let ? be a Galois ring of characteristic 4,X a cyclotomic scheme over ? with classD and ? the Terwilliger algebra ofX, then we show that most of the irreducible ?-modules have standard forms; otherwise, certain relations of the Jacobi sums hold. When the classD is three, we can completely determine the irreducible ?-modules using Jacobi sums.

On the pure Jacobi sums

where χ, ψ is a non trivial character of Fq , whose value at 0 is defined to be 0. It is well known that the absolute value of J(χ, ψ) is √ q = p, when χψ is not principal. According to [11], [9], call the Jacobi sum J(χ, ψ) pure if J(χ, ψ)/p is a root of unity. Let ord(χ) be the order of χ in F×q . From now on in this paper, we assume that ord(ψ) = 2 and ord(χ) = n ≥ 3. This special type of Jacobi sums play an important role in evaluating the argument of Gauss sum:

On Jacobi Sums, Multinomial Coefficients, and p-adic Hypergeometric Functions

We extend the methods of our previous article to express special values of p-adic hypergeometric functions in terms of the p-adic gamma function and Jacobi sums over general finite fields. These results are obtained via p-adic congruences for Jacobi sums in terms of multinomial coefficients, and allow one to more fully exploit classical hypergeometric identities to obtain p-adic unit root formulae.

On the Number of Zeros of Diagonal Forms

In this paper, we study the number of zeros of the equation $X_1^e + X_2^e + \cdots + X_n^e \equiv 0\; \pmod p$, where $e > 1$ is a positive integer and $p \equiv 1\; \pmod e$ is a prime. For $e = 5,7$, we compute the recurrence satisfied by these numbers, using the generalized Jacobi sums.

On Shioda's problem about Jacobi sums

In the present paper, we will give a positive result relating to the l-part of Shioda’s problem [2] on Jacobi sums J (a) l (p) under a certain condition (see Corollary to Theorem 2 of the present paper), as an application of our congruence for Jacobi sums [1, Theorem 2] (see also Theorem 1 of the present paper). Let l be any prime number such that l ≥ 5, and let ζl be a primitive lth root of unity in C (the field of complex numbers). Let Q be the field of rational numbers and let Z be the ring of rational integers. Put k = Q(ζl). For any integer r ≥ 1 and any a = (a1, . . . , ar) ∈ Z and for any prime ideal p of k which is prime to l, let

On Shioda's problem about Jacobi sums II

In the present paper, we will give a complete affirmative answer to the l-part of Shioda’s problem ([5, Question 3.4]) on Jacobi sums J (a) l (p), and to the conjecture (F. Gouvea and N. Yui [1, Conjecture (1.9)]) which comes from Shioda’s problem and my congruences for Jacobi sums (see [3, Theorem 2]) (see Theorem 1 and its Corollary of the present paper). We retain the notation of [4], but l is any odd prime number here. Furthermore, let n be any positive integer and let ζm be a primitive mth root of unity in C for any positive integer m. Let Q be the algebraic closure of Q in C. We fix an algebraic closure Ql of Ql, and by a fixed imbedding Q ↪→ Ql we consider Q as a subfield of Ql. Let M be any finite unramified extension of Ql in Ql, and put Mn = M(ζln) and πn = ζln − 1. Then πn is a prime element of Mn. Let σ−1 ∈ G = Gal(Mn/M) (the Galois group of Mn over M) be such that ζ σ−1 ln = ζ −1 ln . Let ordMn denote the normalized additive valuation of Mn, and let Un = U(Mn) be the group of principal units in Mn: Un = U(Mn) = {x ∈Mn | ordMn(x− 1) ≥ 1}.

Power Residue Character of Jacobi Sums

In two previous papers [ Proc. Amer. Math. Soc. 117 (1993), 877- 884], [ J. Number Theory 44 (1993), 214-221], a reciprocity relation for the power residue symbol of odd prime exponent, between Jacobi sums, was conjectured then proved. This is here extended to the case of an arbitrary exponent, as a consequence of an expression for the power residue character of a Jacobi sum, modulo a rational prime power, in terms of Fermat quotients.

On the Jacobi Sums for Finite Commutative Rings with Identity

In a previous paper [J. Number Theory, 39 (1991), 50-64], we obtained a basic relationship between Gauss sums and Jacobi sums for a ring of residues modulo a prime power, and determined the absolute value of the Jacobi sums. This paper generalizes those results to finite commutative rings with identity.

A Note On the Norms of Algebraic Numbers Associated to Jacobi Sums

A Note On the Norms of Algebraic Numbers Associated to Jacobi Sums

Shtukas and Jacobi sums

We show how the analogues of Jacobi sums, in the context of function fields, introduced and studied in [T1, T2, T3] can be obtained from shtukas introduced and studied in [D2, D3, M]. We apply this to obtain some results on the prime factorization of analogues of Gauss sums and to prove an analogue of the Gross-Koblitz formula for general function field, generalizing the results in [T2]. For this purpose, we also introduce and interpolate a new analogue of gamma function.

Barnes' First Lemma and its Finite Analogue

AbstractWe give a parallel proof of Barnes' first lemma and of its finite analogue. In both cases we use the Mellin transform. In the classical case, the proof avoids the residue theorem. In the finite case the Gamma function is replaced by the Gaussian sum function and the beta function by the Jacobi sum function.

Reciprocity for Jacobi Sums

Let l be an odd prime number and p, q be two prime numbers ≡ 1 (mod l). If χ, χ′ (resp. ψ, ψ′) are multiplicative characters of order l on the finite field Fp (resp. Fq), then the corresponding Jacobi sums J(χ, χ′) and J(ψ, ψ′) satisfy the reciprocity relation [formula] where (−) denotes the lth power residue symbol over the cyclotomic field of lth roots of unity.

Quintic Reciprocity

An expression for the rational inversion factor of the power residue symbol, of odd prime exponent $n \equiv 1\;(\bmod 4)$ (mod 4), is given. It is applied to the quintic case, where the resulting expression involves only a rational quadratic form representation of primes and the power residue character of Jacobi sums. A reciprocity relation for Jacobi sums is then deduced, for $n = 5$, and conjectured to hold for all odd prime exponents $n$.

A note on Jacobi sums, III

A note on Jacobi sums, III

On the conductor of 2-adic Hilbert norm residue symbols

For a fixed prime p, let ζ n be a primitive p n th root of 1 in some algebraic closure of Q p . Let L n = Q p ( ζ n ), and let ( , ) n be the p n th Hilbert norm residue symbol of L n ∗. We compute here the conductor of the character of L n ∗, defined by (, a) n , in the case where p = 2, and a ϵ Q 2 ∗. We then calculate the conductor of the local component of Jacobi sum Hecke characters, ramified only at 2. This is an easy application of the first part, since in [6], we have expressed such characters in terms of Hilbert symbols. One could now use these results to compute the root numbers for the functional equation of the Hecke L-functions associated with these characters. Coleman and McCallum have obtained formulas for the conductor of the norm residue symbol of the p n th cyclotomic field for odd p and for the conductors of Jacobi sum Hecke characters unramified at 2. The computation of the root numbers, in the case of p odd, was carried out by Rohrlich in [8].

Apéry numbers, Jacobi sums, and special values of generalized p-adic hypergeometric functions

We present some summation formulae for some special values of ratios of generalized p-adic hypergeometric functions in terms of the p-adic gamma function. By well-known methods such formulae yield expressions for roots of congruence zeta-functions in terms of Jacobi sums in the nonsingular case. We also express various formal-group congruences, including those involving Apéry numbers, in terms of p-adic hypergeometric functions at singular and nonsingular points of the associated differential equation. These congruences yield p-adic analytic formulae for unit roots of certain Hecke polynomials.

Supplementary difference sets and Jacobi sums

Let q=e∝+1 be an odd prime power and Ci, 1⩽i⩽e-1, be cyclotomic classes of the eth power residues in F=GF(q). Let Ai with #Ai=ui, 1⩽i⩽n, be non-empty subsets of Ω={0,1,…,e−1} and let Di=∪lϵAiCl, 1⩽i⩽n. Here we prove that D1,…, Dn become n−{q:u1∝, u2∝, …, un∝;λ} supplementary difference sets if and only if the following equations are satisfied: 1.(i) Σni=1ui(ui∝−1)≡0 (mode), (ii) Σni=1Σe−1m=0π(χm, χ−t)ωi,mωi,t−m=0, for all t, 1 ⩽t⩽e−1, where π(χm, χ−t) is the Jacobi sum for the eth power residue characters and ωi,m=ΣlϵAiζ−lme, where ζe is a p rimitive eth root of unity. Furthermore, we give numerical results for e=2,n=1,2 and for e=4,n=1, 2, 3, 4.

On the relation between units and Jacobi sums in prime cyclotomic fields

On the relation between units and Jacobi sums in prime cyclotomic fields

On the Jacobi sums modulo Pn

In this paper we study the Jacobi sums over a ring of residues modulo a prime power and obtain the basic relationship between Gauss sums and Jacobi sums, which allows us to determine the absolute value of the Jacobi sums. As an application, we discuss the congruence x 1 p + … + x r p ≡ 1 (mod p 2), where p is an odd prime.

On Jacobi sum Hecke characters ramified only at 2

Let K be the cyclotomic field of the mth roots of unity in some fixed algebraic closure of Q. Weil has shown in [Jacobi sums as Großencharaktere, Trans. Amer. Math. Soc. 73 (1952), 487–495] that Jacobi sums induce Hecke characters on K modulo m 2, or equivalently homomorphisms of the idèle class group of K into K ∗ . Given such a Hecke character ϱ, we then obtain for every place p of K a continuous homomorphism ϱ p (the local component of ϱ at p), from ( K p ) ∗ into K ∗ . When p does not divide m, ϱ p is completely determined. When p divides m the determination of ϱ p is much more difficult. Coleman and McCallum in [Stable reduction of Fermat curves and Jacobi sum Hecke characters, J. Reine Angew. Math. 385 (1988), 41–101] have given formulas and computed the conductors of the local components of ϱ at p for p∥ m and m odd. Hasse, in [Zetafunktion und L-Funktionen zu einem arithmetischen Funktionenkorper vom Fermatschen Typus, Abh. Deut. Akad. Wiss. Berlin Kl. Math. Nat. 1954 4 (1955)] has computed the local component and the conductor when m is prime, while in [C. Jensen, Uber die Führer einer Klasse Heckescher Großencharaktere, Math. Scand. 8 (1960), 81–96; D. Rohrlich, Jacobi sums and explicit reciprocity laws, Compositio Math. 60 (1986), 97–110; C.-G. Schmidt, Uber die Führer von Gauss'schen Summen als Großencharaktere, J. Number Theory 12, No. 3 (1980), 283–310] the authors have given estimates and in some cases have determined the conductor of the local component. Here we deal with the case where m = 2 n , and we give an explicit formula for the local component in terms of Hilbert symbols, similar to the one in [R. Coleman and W. McCallum, Stable reduction of Fermat curves and Jacobi sum Hecke characters, J. Reine Angew. Math. 385 (1988), 41–101]. In [D. Prapavessi, On the conductor of 2-adic Hilbert norm residue symbols, to appear in J. Algebra] we use this result to compute the conductor of ϱ when m = 2 n .