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Let L d be the Lattes map associated to the multiplication-by-d endomorphism of an elliptic curve E defined over a finite field F q . We determine the density δ ( L d , q ) of periodic points for L d in P 1 ( F q ) . We show that the periodic point densities δ ( L d , q n ) converge as n → ∞ along certain arithmetic progressions, and compute simple explicit formulas for δ ( L l , q ) when l is a prime and E belongs to a special family of supersingular elliptic curves.

In this paper we obtain explicit estimates and existence results on the number of F q -rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the geometric properties of the varieties defined by the systems involved. We apply these results to a generalization of Waring's problem and the distribution of solutions of congruences modulo a prime number.

We provide a number of new conjectures and questions concerning the syzygies of P 1 × P 1 . The conjectures are based on computing the graded Betti tables and related data for large number of different embeddings of P 1 × P 1 . These computations utilize linear algebra over finite fields and high-performance computing.

In the area of symbolic-numerical computation within computer algebra, an interesting question is how “close” a random input is to the “critical” ones. Examples are the singular matrices in linear algebra or the polynomials with multiple roots for Newton's root-finding method. Bounds, sometimes very precise, are known for the volumes over R or C of such neighborhoods of the varieties of “critical” inputs; see the references below. This paper deals with the discrete version of this question: over a finite field, how many points lie in a certain type of neighborhood around a given variety? A trivial upper bound on this number is given by the product (size of the variety) ⋅ (size of a neighborhood of a point). It turns out that this bound is usually asymptotically tight, in particular for the singular matrices, polynomials with multiple roots, and pairs of non-coprime polynomials. The interesting question then is: for which varieties is this bound not asymptotically tight? We show that these are precisely those that admit a shift, that is, where one absolutely irreducible component of maximal dimension is a shift (translation by a fixed nonzero point) of another such component. Furthermore, the shift-invariant absolutely irreducible varieties are characterized as being cylinders over some base variety. Computationally, determining whether a given variety is shift-invariant turns out to be intractable, namely NP-hard even in simple cases.

In this paper k always denotes a finite field of characteristic p > 0 . A variety over k means a reduced scheme separated and of finite type over k. An etale cover is always assumed to be finite (as in [SGA1, I, Def. 4.9] ).

We characterize the unit group of semisimple group algebras $\mathbb{F}_qG$ of some non-metabelian groups, where $F_q$ is a field with $q=p^k$ elements for $p$ prime and a positive integer $k$. In particular, we consider all 6 non-metabelian groups of order 48, the only non-metabelian group $((C_3\times C_3)\rtimes C_3)\rtimes C_2$ of order 54, and 7 non-metabelian groups of order 72. This completes the study of unit groups of semisimple group algebras for groups upto order 72.

We extend the result of Bisson [4] to compute the endomorphism rings of principally polarized, absolutely simple, and ordinary abelian surfaces defined over finite fields in subexponential time in the size of the base field. The abelian surfaces covered here were excluded in [4] . This is accomplished by using techniques introduced in [17] to efficiently determine overorders for Bass and Gorenstein orders. In addition we show that the endomorphism rings of certain principally polarized, absolutely simple, and ordinary abelian surfaces are not computable in subexponential time with current methods [4] , [21] , including ours.

Recursive MDS matrices are used for the design of linear diffusion layers in lightweight cryptographic applications. Most of the works on the construction of recursive MDS matrices either consider matrices over finite fields or block matrices over G L ( m , F 2 ) . In the first case, there have been works on the direct construction of recursive MDS matrices. The latter case is hard to deal with because of its non-commutative nature. There has not been any serious attempt to look for recursive MDS matrices over finite commutative rings, in particular over local rings of even characteristic. In this work, we present several methods for the construction of recursive MDS companion matrices over finite commutative rings. The main tools are the simple expressions for the determinant of (generalized) Vandermonde and linearized matrices. We show that the determinant of a linearized matrix over a finite commutative ring of prime characteristic can be expressed in a simple form. We discuss a technique called subring construction with which MDS matrices over product rings can be constructed using MDS matrices over subrings. We give a few examples of recursive MDS companion matrices over local rings of even characteristic. We also discuss some results on the nonexistence of recursive MDS matrices over certain rings for some parameter choices.

Let 𝔽q[ɛ] := 𝔽q [X]/(X4 − X3) be a finite quotient ring where ɛ4 = ɛ3, with 𝔽q is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve over 𝔽q[ɛ], ɛ4 = ɛ3 of characteristic p ≠ 2, 3 given by homogeneous Weierstrass equation of the form Y 2Z = X3 + aXZ2 + bZ3 where a and b are parameters taken in 𝔽q[ɛ]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve Ea,b(𝔽q[ɛ]) and we will show that Eπ0(a),π0(b)(𝔽q) and Eπ1(a),π1(b)(𝔽q) are two elliptic curves over the finite field 𝔽q, such that π0 is a canonical projection and π1 is a sum projection of coordinate of element in 𝔽q[ɛ]. Precisely, we give a classification of elements in elliptic curve over the finite ring 𝔽q[ɛ].

Let $\mathbb F$ be a finite field. Consider a direct sum $V$ of an infinite number of copies of $\mathbb F$, consider the dual space $V^\diamond$, i.~e., the direct product of an infinite number of copies of $\mathbb F$. Consider the direct sum ${\mathbb V}=V\oplus V^\diamond$. The object of the paper is the group $\mathbf{GL}$ of continuous linear operators in $\mathbb V$. We reduce the theory of unitary representations of $\mathbf{GL}$ to projective representations of a certain category whose morphisms are linear relations in finite-dimensional linear spaces over $\mathbb F$. In fact we consider a certain family $ Q_\alpha$ of subgroups in $\mathbb V$ preserving two-element flags, show that there is a natural multiplication on spaces of double cosets with respect to $ Q_\alpha$, and reduce this multiplication to products of linear relations. We show that this group has type $\mathrm{I}$ and obtain an 'upper estimate' of the set of all irreducible unitary representations of $\mathbf{GL}$., Comment: 48pp, a revised version

In this paper, we give a characterization of finite-dimensional nilpotent Lie algebras of breadth 3 over finite fields of odd characteristic. This characterization parallels to the one for finite p-groups of breadth 3 given earlier in [5] .

It is proved that with finitely many possible exceptions, each cyclotomic scheme over a finite field is determined up to isomorphism by the tensor of 2 2 -dimensional intersection numbers; for infinitely many schemes, this result cannot be improved. As a consequence, the Weisfeiler–Leman dimension of a Paley graph or tournament is at most 3 3 with possible exception of several small graphs.

Let Fd be the finite field with d elements. Extraordinary subgroups in Fd × Fd play an important role in the field of quantum information theory, especially for the study of mutually unbiased bases. Recently, Ghiu et al. introduced the concept of supersquare of order d which is related to extraordinary subgroups. They have given a method of construction of the mutually orthogonal supersquares, and determined all the complete sets of mutually orthogonal extraordinary supersquares of order 4. In this article, we present the construction of a complete set of mutually orthogonal extraordinary supersquares of order pn where p is a prime. We also determine all the complete sets of mutually orthogonal extraordinary supersquares of order 9.

We introduce a model for random chain complexes over a finite field. The randomness in our complex comes from choosing the entries in the matrices that represent the boundary maps uniformly over $\mathbb{F}_q$, conditioned on ensuring that the composition of consecutive boundary maps is the zero map. We then investigate the combinatorial and homological properties of this random chain complex.

The known algorithms for synthesizing irreducible polynomials have a significant drawback: their computational complexity, as a rule, exceeds the quadratic one. Moreover, consequently, as a consequence, the construction of large-degree polynomials can be implemented only on computing systems with very high performance. The proposed algorithm is base on the use of so-called fiducial grids (ladders). At each rung of the ladder, simple recurrent modular computations are performers. The purpose of the calculations is to test the irreducibility of polynomials over Galois fields of arbitrary characteristics. The number of testing steps coincides with the degree of the synthesized polynomials. Upon completion of testing, the polynomial is classifieds as either irreducible or composite. If the degree of the synthesized polynomials is small (no more than two dozen), the formation of a set of tested polynomials is carried out using the exhaustive search method. For large values of the degree, the test polynomials are generating by statistical modeling. The developed algorithm allows one to synthesize binary irreducible polynomials up to 2Kbit on personal computers of average performance

We show that some types for supercuspidal representations of tamely ramified $p$-adic groups that appear in Jiu-Kang Yu's work are geometrizable. To do so, we define a function-sheaf dictionary for one-dimensional characters of arbitrary smooth group schemes over finite fields. In previous work we considered the case of commutative smooth group schemes and found that the standard definition of character sheaves produced a dictionary with a nontrivial kernel. In this paper we give a modification of the category of character sheaves that remedies this defect, and is also extensible to non-commutative groups. We then use these commutative character sheaves to geometrize the linear characters that appear in the types introduced by Jiu-Kang Yu, assuming that the character vanishes on a certain derived subgroup. To define geometric types, we combine commutative character sheaves with Gurevich and Hadani's geometrization of the Weil representation and Lusztig's character sheaves., Updated to fix problem with characters not vanishing on derived subgroup. 27 pages

We use the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface to show the existence of a regular surface with canonical map of degree 36 and of an irregular surface with canonical map of degree 27. As a by-product, we get equations (over a finite field) for the $\mathbb Z/3$-invariant fibres of the Albanese fibration of the Cartwright-Steger surface and show that they are smooth., Ancillary files with Magma code included. Final version, to appear in J Algebraic Geom

Inspired by the classical setting, Goss defined the $L$-series of Drinfeld $A$-modules corresponding to representations of the absolute Galois group of a rational function field. In this paper, for a given Drinfeld $A$-module $\phi$ of rank 2 defined over the finite field $\mathbb{F}_q$, we give explicit formulas for the values of Goss $L$-series at positive integers $n$ such that $2n+1\leq q$ in terms of polylogarithms and coefficients of the logarithm series of $\phi$.

Let G G be a linear algebraic group over a field k k of characteristic 0. We show that any two connected semisimple k k -subgroups of G G that are conjugate over an algebraic closure of k k are actually conjugate over a finite field extension of k k of degree bounded independently of the subgroups. Moreover, if k k is a real number field, we show that any two connected semisimple k k -subgroups of G G that are conjugate over the field of real numbers R {\mathbb {R}} are actually conjugate over a finite real extension of k k of degree bounded independently of the subgroups.

We consider a special class of idempotent of semisimple group algebras which we call essential. We give some criteria to decide when a primitive idempotent is essential; then we consider group algebras of cyclic group over finite fields, establish the number of essential idempotents in this case and find a relation among essential idempotents in different algebras. Finally we apply this ideas to coding theory and compute examples of codes with the best known weight.

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields.

In this paper, we completely determine the slopes and weights of the L-functions of an important class of exponential sums arising from analytic number theory. Our main tools include Adolphson-Sperber's work on toric exponential sums and Wan's decomposition theorems. One consequence of our main result is a sharp estimate of these exponential sums. Another consequence is to obtain an explicit counterexample of Adolphson-Sperber's conjecture on weights of toric exponential sums.

In this paper, we prove certain multiplicity one theorems and define twisted gamma factors for irreducible generic cuspidal representations of split $$\mathrm {G}_2$$ over finite fields k of odd characteristic. Then we prove the first converse theorem for exceptional groups, namely, $${\mathrm {GL}}_1$$ and $${\mathrm {GL}}_2$$ -twisted gamma factors will uniquely determine an irreducible generic cuspidal representation of $${\mathrm {G}}_2(k)$$ .

The aim of this paper is to study the existence of nontrivial, nonsingular zeros of a nonhomogeneous polynomial defined over a finite field. To accomplish this, we determine conditions that guarant...

This is an expository paper on algebraic aspects of exponential sums over finite fields. This is a new direction. Various examples, results and open problems are presented along the way, with particular emphasis on Gauss periods, Kloosterman sums and one variable exponential sums. One main tool is the applications of various p-adic methods. For this reason, the author has also included a brief exposition of certain p-adic estimates of exponential sums. The material is based on the lectures given at the 2020 online number theory summer school held at Xiamen University. Notes were taken by Shaoshi Chen and Ruichen Xu.

This research aims to give a splitting structure of the projective line over the finite field of order twenty-seven that can be found depending on the factors of the line order. Also, the line was partitioned by orbits using the companion matrix. Finally, we showed the number of projectively inequivalent -arcs on the conic through the standard frame of the plane PG(1,27)

Let F q denote the finite field with q elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over F q in terms of the number of rational points on elliptic curves. In the case where q is a prime number, we give a way to calculate these numbers. As a consequence of these results, we characterize maximal and minimal curves given by equations of the forms a x 3 + b y 3 + c z 3 = 0 and a x 4 + b y 4 + c z 4 = 0 .

Let 𝔽 q {\mathbb{F}_{q}} be the finite field of order q, where q is an odd prime power. Then a k-dimensional quadratic subspace ( W , Q ) {(W,Q)} of ( 𝔽 q n , x 1 2 + x 2 2 + ⋯ + x n 2 ) {(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})} is called dot 𝐤 {\operatorname{dot}_{\mathbf{k}}} -subspace if Q is isometrically isomorphic to x 1 2 + x 2 2 + ⋯ + x k 2 {x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}} . In this paper, we obtain bounds for the number of incidences I ⁢ ( 𝒦 , ℋ ) {I(\mathcal{K},\mathcal{H})} between a collection 𝒦 {\mathcal{K}} of dot k {\operatorname{dot}_{k}} -subspaces and a collection ℋ {\mathcal{H}} of dot h {\operatorname{dot}_{h}} -subspaces when h ≥ 4 ⁢ k - 4 {h\geq 4k-4} , which is given by | I ⁢ ( 𝒦 , ℋ ) - | 𝒦 | ⁢ | ℋ | q k ⁢ ( n - h ) | ≲ q k ⁢ ( 2 ⁢ h - n - 2 ⁢ k + 4 ) + h ⁢ ( n - h - 1 ) - 2 2 ⁢ | 𝒦 | ⁢ | ℋ | . \Bigl{\lvert}I(\mathcal{K},\mathcal{H})-\frac{\lvert\mathcal{K}\rvert\lvert% \mathcal{H}\rvert}{q^{k(n-h)}}\Bigr{\rvert}\lesssim q^{\frac{k(2h-n-2k+4)+h(n-% h-1)-2}{2}}\sqrt{\lvert\mathcal{K}\rvert\lvert\mathcal{H}\rvert}. In particular, we improve the error term in [N. D. Phuong, P. V. Thang and L. A. Vinh, Incidences between planes over finite fields, Proc. Amer. Math. Soc. 147 2019, 5, 2185–2196] obtained by Phuong, Thang and Vinh for general collections of affine subspaces in the presence of our additional conditions.

The goal of this paper is to prove theorems that elucidate the Cohen-Lenstra-Martinet conjectures for the distributions of class groups of number fields, and further the understanding of their implications. We start by giving a simpler statement of the conjectures. We show that the probabilities that arise are inversely proportional the to number of automorphisms of structures slightly larger than the class groups. We find the moments of the Cohen-Lenstra-Martinet distributions and prove that the distributions are determined by their moments. In order to apply these conjectures to class groups of non-Galois fields, we prove a new theorem on the capitulation kernel (of ideal classes that become trivial in a larger field) to relate the class groups of non-Galois fields to the class groups of Galois fields. We then construct an integral model of the Hecke algebra of a finite group, show that it acts naturally on class groups of non-Galois fields, and prove that the Cohen-Lenstra-Martinet conjectures predict a distribution for class groups of non-Galois fields that involves the inverse of the number of automorphisms of the class group as a Hecke-module.

Let $$D_n(x;a)$$ and $$E_n(x;a)\in {\mathbb {F}}_q[x]$$ be Dickson polynomials of first and second kind respectively, where $${\mathbb {F}}_q$$ is a finite field with q elements. In this article we show explicitly the irreducible factors of these polynomials in the case that every prime divisor of n divides $$q-1$$ . This result generalizes the results found in (Finite Fields Appl. 3:84–96, 1997; Explicit factorization of cyclotomic and Dickson polynomials over finite fields, Springer, Berlin, 2007; Finite Fields Appl. 38:40-56, 2016; Discrete Math. 342:111618, 2019).

We calculate characters of Weil representations of unitary groups over finite fields geometrically. As a byproduct, we deduce another proof of Gerardin’s trace formula for the Weil representation. ...

We study the number of rational points of smooth projective curves over finite fields in some relative situations in the spirit of a previous paper [HP19] from an euclidean point of vue. We prove some kinds of relative Weil bounds, derived from Schwarz inequality for some "relative parts" of the diagonal and of the graph of the Frobenius on some euclidean sub-spaces of the numerical space of the squared curve endowed with the opposite of the intersection product.

We give a structure theorem for the $m$-torsion of the Jacobian of a general superelliptic curve $y^m=F(x)$. We study existence of torsion on curves of the form $y^q=x^p-x+a$ over finite fields of characteristic $p$. We apply those results to bound from below the Mordell-Weil ranks of Jacobians of certain superelliptic curves over $\mathbb Q$., Comment: 11 pages

We define additive skew G-codes over finite fields and discuss several dualities attached to these codes. We examine the properties of self-dual skew G-codes and in particular we show that the dual, under any duality, of an additive skew G-code is also an additive skew G-code. Additionally, we propose a matrix construction for additive skew G-codes and use it to construct several examples of extremal self-dual additive skew G-codes over the finite field $${\mathbb {F}}_4$$ . Such codes have a strong connection to quantum error correcting codes.

In this article we find finite field analogues of certain transformations satisfied by the classical hypergeometric series. Using properties of Gauss and Jacobi sums we evaluate certain character sums to establish these transformations. We then use these transformations to evaluate explicitly several special values of $${_2}F_1$$ hypergeometric functions over finite fields. Certain special values of $${_2}F_1$$ hypergeometric functions over finite fields containing trivial and quadratic characters obtained by Ono follow from our special values of $${_2}F_1$$ hypergeometric functions containing arbitrary characters as parameters. One of the finite field analogues of algebraic hypergeometric identities given by Fuselier, Long, Ramakrishna, Swisher, and Tu also follows from one of our transformation formulas.

It is well-known that each left ideal in a matrix rings over a finite field is generated by an idempotent matrix. In this work we compute the number of left ideals in these rings, the number of different idempotents generating each left ideal, and give explicitly a set of idempotent generators of all left ideals of a given rank. We then apply these results to give examples of left group codes that have best possible minimum weight.

The reduced anisotropic unitary Whitehead groups of henselian division algebras with unitary involutions are computed in the cases where the centers of residue algebras are of special types.

In this paper, we investigate an inverse problem on the Gauss sums of characters of finite fields. Namely, given a nontrivial additive character ψ, for two multiplicative characters χ 1 and χ 2 of F q n × , if the Gauss sums G ( χ 1 ⋅ η , ψ ) = G ( χ 2 ⋅ η , ψ ) for all characters η of F q × , when is χ 1 equal to χ 2 up to a Frobenius twist? This paper proves that the answer is positive for regular characters when n ≤ 5 , or for n q − 1 2 q + 1 in the appendix by Zhiwei Yun. In addition, we conjectured that the answer is positive when n is prime.

In literature, there are several methods to construct Gold sequences. One of the constructions is via the trace function from extension field of $\mathbb {F}_{2}$ . Estimation of correlation of this construction is based on number of rational points of elliptic curves. In this article, we generalize this construction from finite fields of even characteristic to odd characteristics by using multiplicative quadratic character. Again, estimation of correlation of this construction is based on number of rational points of elliptic curves. Thus, we obtain binary sequences which have more flexibility on length while still possessing low correlation property. Moreover, some of the sequences are optimally balanced.

In this paper, we introduce a new class of additive codes over finite fields, viz. multi-twisted (MT) additive codes, which are generalizations of constacyclic additive codes. We study their algebraic structures by writing a canonical form decomposition and provide an enumeration formula for these codes. By placing ordinary, Hermitian and $$*$$ trace bilinear forms, we further study their dual codes and derive necessary and sufficient conditions under which a MT additive code is self-dual and self-orthogonal. We also derive a necessary and sufficient condition for the existence of a self-dual MT additive code over a finite field, and provide enumeration formulae for all self-dual and self-orthogonal MT additive codes over finite fields with respect to the aforementioned trace bilinear forms. We also obtain several good codes within the family of MT additive codes over finite fields.

Let Fq be a finite field and Fq((X−1 )) the field of formal power series with coefficients in Fq. The purpose of this paper is to exhibit a family of transcendental continued fractions of formal power series over a finite field through some specific irregularities of its partial quotients

The (p-primary) equivariant Euler characteristics of the buildings for the general unitary groups over finite fields are determined.

It is well known that any finite commutative ring is isomorphic to a direct product of local rings via the Chinese remainder theorem. Hence, there is a great significance to the study of character sums over local rings. Character sums over finite rings have applications that are analogous to the applications of character sums over finite fields. In particular, character sums over local rings have many applications in algebraic coding theory. In this paper, we firstly present an explicit description on additive characters and multiplicative characters over a certain local ring. Then we study Gaussian sums, hyper Eisenstein sums and Jacobi sums over a certain local ring and explore their properties. It is worth mentioning that we are the first to define Eisenstein sums and Jacobi sums over this local ring. Moreover, we present a connection between hyper Eisenstein sums over this local ring and Gaussian sums over finite fields, which allows us to give the absolute value of hyper Eisenstein sums over this local ring. As an application, several classes of codebooks with new parameters are presented.

Let k be a finite field of characteristic p, and G a compact p-adic analytic group. Write kG for the completed group ring of G over k. In this paper, we describe the structure of the ring kG/P, where P is a minimal prime ideal of kG. We give an explicit isomorphism between kG/P and a matrix ring with coefficients in the ring ${(k'G')_\alpha }$ , where $k'/k$ is a finite field extension, $G'$ is a large subquotient of G with no finite normal subgroups, and (–)α is a “twisting” operation that preserves many desirable properties of the ring structure. We demonstrate the usefulness of this isomorphism by studying the correspondence induced between certain ideals of kG and those of ${(k'G')_\alpha }$ , and showing that this preserves many useful “group-theoretic” properties of ideals, in particular almost-faithfulness and control by a closed normal subgroup.

We consider the subgroups $ H $ in a symplectic or orthogonal group over a finite field of odd characteristic such that $ O_{r}(H)\neq 1 $ for some odd prime $ r $ . We obtain a refinement of the well-known Aschbacher Theorem on subgroups of classical groups for this case.