Stickelberger's Theorem Research Articles

Gauss sums and the maximum cliques in generalized Paley graphs of square order

Let $GP(q,d)$ be the $d$-Paley graph defined on the finite field $\mathbb{F}_q$. It is notoriously difficult to improve the trivial upper bound $\sqrt{q}$ on the clique number of $GP(q,d)$. In this paper, we investigate the connection between Gauss sums over a finite field and the maximum cliques of their corresponding generalized Paley graphs. We show that the trivial upper bound on the clique number of $GP(q,d)$ is tight if and only if $d \mid (\sqrt{q}+1)$, which strengthens the previous related results by Broere-D\"oman-Ridley and Schneider-Silva. We also obtain a new simple proof of Stickelberger's theorem on evaluating semi-primitive Gauss sums.

Gauss sums, Stickelberger’s theorem and the Gras conjecture for ray class groups

Let $k$ be a real abelian number field and $p$ an odd prime not dividing $[k:\mathbb{Q}]$. For a natural number $d$, let $E_d$ denote the group of units of $k$ congruent to $1$ modulo $d$, $C_d$ the subgroup of $d$-circular units of $E_d$, and $\mathfrak{C}(d)$ the ray class group of modulus $d$. Let $\rho$ be an irreducible character of $G=\mathrm{Gal}(k/\mathbb{Q})$ over $\mathbb{Q}_p$ and $e_{\rho} \in \mathbb{Z}_p[G]$ the corresponding idempotent. We show that if the ramification index of $p$ in $k$ is less than $p-1$, then $|e_{\rho} \mathrm{Syl}_p(E_d/C_d) | = |e_{\rho} \mathrm{Syl}_p(\mathfrak{C}_d)|$ where $\mathfrak{C}_d$ is the part of $\mathfrak{C}(d)$ where $G$ acts non-trivially. This is a ray class version of the Gras Conjecture. In the case when $p \mid [k:\mathbb{Q}]$, similar but slightly less precise results are obtained. In particular, beginning with what could be considered a Gauss sum for real fields, we construct explicit Galois annihilators of $\mathrm{Syl}_p(\mathfrak{C}_{\mathfrak{a}})$ akin to the classical Stickelberger Theorem.

A small remark on the filtered $\varphi$-module of Fermat varieties and Stickelberger's theorem

We show that the weakly admissibility of the filtered $\varphi$-module with coefficients of Fermat varieties in the sense of Fontaine essentially expresses Stickelberger's theorem in Iwasawa theory. In particular, it gives us a simple re-proof of the weakly admissibility of it.

The Proof of Lin’s Conjecture via the Decimation-Hadamard Transform

In 1998, Lin presented a conjecture on a class of ternary sequences with ideal two-level autocorrelation. Those sequences have a very simple structure, i.e., their trace representation has two trace monomial terms. In this paper, we present a proof for the conjecture. The mathematical tools employed are the second-order multiplexing decimation-Hadamard transform, Stickelberger's theorem, the Teichmuller character, and combinatorial techniques for enumerating the Hamming weights of ternary numbers. As a by-product, we also prove that the ternary sequences conjectured by Lin are Hadamard equivalent to ternary m-sequences.

Some new maps and ideals in classical Iwasawa theory with applications

We introduce a new ideal {\mathfrak D} of the p-adic Galois group-ring associated to a real abelian field and a related ideal {\mathfrak J} for imaginary abelian fields. Both result from an equivariant, Kummer-type pairing applied to Stark units in a Z_p-tower of abelian fields and {\mathfrak J} is linked by explicit reciprocity to a third ideal {\mathfrak S} studied more generally in a previous work. This leads to a new and unifying framework for the Iwasawa Theory of such fields including a real analogue of Stickelberger's Theorem, links with certain Fitting ideals and \Lambda-torsion submodules, and a new exact sequence related to the Main Conjecture.

On the Dual of Certain Ternary Weakly Regular Bent Functions

In 2006, Helleseth and Kholosha conjectured and partially proved the existence of a class of ternary weakly regular monomial bent functions and also the expression for the dual bent function up to the sign value. The bentness was finally proved later in 2009 using a complicated technique that employs Stickelberger's theorem. Extensively using the previously found results and approaches, in this paper, a surprisingly short proof for the conjectured expression of the dual is given but without resolving the sign ambiguity. Furthermore, we resolve the sign by finding the trace representation of the dual function.

Binary Kloosterman Sums Modulo 256 and Coefficients of the Characteristic Polynomial

Kloosterman sums are exponential sums on finite fields that have important applications in cryptography and coding theory. We use Stickelberger's theorem and the Gross-Koblitz formula to determine the value of the binary Kloosterman sum at <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$a$</tex></formula> modulo 64, modulo 128, and modulo 256 in terms of coefficients of the characteristic polynomial of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$a$</tex> </formula> .

Binary Kloosterman sums using Stickelberger's theorem and the Gross–Koblitz formula

Binary Kloosterman sums using Stickelberger's theorem and the Gross–Koblitz formula

A non-abelian Stickelberger theorem

AbstractLet L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring ℤ(p)[G] that annihilates the p-part of the class group of L.

Parity of the number of irreducible factors for composite polynomials

Various results on the parity of the number of irreducible factors of given polynomials over finite fields have been obtained in the recent literature. Those are mainly based on Stickelberger's and Swan's theorem in which discriminants of polynomials over a finite field or the integral ring Z play an important role. In this paper we consider discriminants of the composition of some polynomials over finite fields. A relation between the discriminants of the composed polynomial and the original ones will be established. We apply this to obtain some results concerning the parity of the number of irreducible factors for several special types of polynomials over finite fields.

Proofs of Two Conjectures on Ternary Weakly Regular Bent Functions

In this paper, we study ternary monomial functions of the form f(x) = Trn(axd), where x isin \BBF 3 n and Trn: \BBF 3 nrarr \BBF 3 is the absolute trace function. Using a lemma of Hou, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising in the 2006 IEEE Transactions on Information Theory paper (vol. 52, pp. 2018-2032, 2006) are weakly regular bent, thus settling a conjecture of Helleseth and Kholosha. We also prove that the Coulter-Matthews bent functions are weakly regular.

Monomial bent functions and Stickelberger's theorem

In this paper we use certain results on the divisibility of Gauss sums, mainly Stickelberger's theorem, to study monomial bent functions. This approach turns out to be especially nice in the Kasami, Gold and Dillon case. As one of our main results we give an alternative proof of bentness in the case of the Kasami exponent. Using the techniques developed here, this proof turns out to be very short and generalizes the previous results by Dillon and Dobbertin to the case where n is divisible by 3. Furthermore, our approach can also be used to deduce properties of the dual function. More precisely, we show that the dual of the Kasami function is not a monomial Boolean function.

Ideal class group annihilators

We study certain correspondences over Drinfeld modular varieties given by sums of Hecke correspondences. We propose generalizations of Stickelberger's theorem for higher dimensions. Using this result, we study anihilators for some cusp forms.

The L-functions of twisted Witt extensions

The L-function of a non-degenerate twisted Witt extension is proved to be a polynomial. Its Newton polygon is proved to lie above the Hodge polygon of that extension. And the Newton polygons of the Gauss–Heilbronn sums are explicitly determined, generalizing the Stickelberger theorem.

Relative $K_{0}$, annihilators, Fitting ideals and Stickelberger phenomena

When $G$ is abelian and $l$ is a prime we show how elements of the relative K-group $K_{0}({\bf Z}_{l}[G], {\bf Q}_{l})$ give rise to annihilator/Fitting ideal relations of certain associated ${\bf Z}[G]$-modules. Examples of this phenomenon are ubiquitous. Particularly, we give examples in which $G$ is the Galois group of an extension of global fields and the resulting annihilator/Fitting ideal relation is closely connected to Stickelberger's Theorem and to the conjectures of Coates and Sinnott, and Brumer. Higher Stickelberger ideals are defined in terms of special values of L-functions; when these vanish we show how to define fractional ideals, generalising the Stickelberger ideals, with similar annihilator properties. The fractional ideal is constructed from the Borel regulator and the leading term in the Taylor series for the L-function. En route, our methods yield new proofs, in the case of abelian number fields, of formulae predicted by Lichtenbaum for the orders of K-groups and etale cohomology groups of rings of algebraic integers.

The theta divisor and the Stickelberger theorem

This paper is devoted to studying certain trivial correspondences provided by theta divisors and their relation to the Brumer-Stark conjecture. © 2005 American Mathematical Society.

A Stickelberger theorem for p-adic Gauss sums

A Stickelberger theorem for p-adic Gauss sums

Higher analogues of Stickelberger's theorem

Let l be an odd prime number, F denote any totally real number field and E/ F be an Abelian CM extension of F of conductor f. In this paper we prove that for every n odd and almost all prime numbers l we have S n(E/F,l)⊂Ann Z l[G(E/F)] H 2( O E[1/l]; Z l(n+1)) where S n ( E/ F, l) is the Stickelberger ideal (Ann. of Math. 135 (1992) 325–360; J. Coates, p-adic L-functions and Iwasawa's theory, in: Algebraic Number Fields by A. Fröhlich, Academic Press, London, 1977). In addition if we assume the Quillen–Lichtenbaum conjecture then S n(E/F,l)⊂Ann Z l[G(E/F)] K 2n( O E) l. To cite this article: G. Banaszak, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Gauss Sums, Jacobi Sums, and p-Ranks of Cyclic Difference Sets

We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are (2 d −1, 2 d−1 −1, 2 d−2 −1) cyclic difference sets in the multiplicative group of the finite field F 2 d of 2 d elements, with d⩾2. We show that, except for a few cases with small d, these difference sets are all pairwise inequivalent. This is accomplished in part by examining their 2-ranks. The 2-ranks of all of these difference sets were previously known, except for those connected with the Segre and Glynn hyperovals. We determine the 2-ranks of the difference sets arising from the Segre and Glynn hyperovals, in the following way. Stickelberger's theorem for Gauss sums is used to reduce the computation of these 2-ranks to a problem of counting certain cyclic binary strings of length d. This counting problem is then solved combinatorially, with the aid of the transfer matrix method. We give further applications of the 2-rank formulas, including the determination of the nonzeros of certain binary cyclic codes, and a criterion in terms of the trace function to decide for which β in F * 2 d the polynomial x 6+ x+ β has a zero in F 2 d , when d is odd.

Minimal Polynomials and Distinctness of Kloosterman Sums

In this paper, we use the Stickelberger theorem on Gauss sums to study the minimal polynomial and distinctness of n-dimensional Kloosterman sums over a finite field Fq of q elements. In particular, we improve some results of Fisher [Contemp. Math. 133 (1992), 81–102; Kloosterman sums as algebraic integers, Math. Ann., to appear.] obtained by ℓ-adic methods. We also show that the minimal polynomials of certain exponential sums (including many Kloosterman sums) are p-Eisensteinian in a broader sense.