Generalized Artin–Mumford curves over finite fields

The Hurwitz curve over a finite field and its Weierstrass points for the morphism of lines

The group is important because it is one of the seven finitely presented isomorphism types of subgroups of the full automorphism group (Aut(Γ3)) of a cubic tree Γ3. These seven groups act arc-transitively on the arcs of Γ3 with a finite vertex stabilizer. In this article, we have parametrized the actions of on the projective line over the finite field, PL(F p ), where p is the Pythagorean prime and has thus shown that there is only one coset diagram depicting the sole conjugacy class of these actions.

Automorphism groups of cyclic p-gonal pseudo-real Riemann surfaces

Let Γ = Cay(G, S), R(G) be the right regular representation of G. The graph Γ is called normal with respect to G, if R(G) is normal in the full automorphism group Aut(Γ) of Γ. Γ is called a bi-normal with respect to G if R(G) is not normal in Aut(Γ), but R(G) contains a subgroup of index 2 which is normal in Aut(Γ). In this paper, we prove that connected tetravalent edge-transitive Cayley graphs on PGL(2, p) are either normal or bi-normal when p ≠ 11 is a prime.

For [Formula: see text] ([Formula: see text]), denote by [Formula: see text] the finite field of order [Formula: see text] and for a positive integer [Formula: see text], let [Formula: see text] be its extension field of degree [Formula: see text]. We establish a sufficient condition for existence of a primitive normal element [Formula: see text] such that [Formula: see text] is a primitive element, where [Formula: see text], with [Formula: see text] satisfying [Formula: see text] in [Formula: see text].

A complete hypergeometric point count formula for Dwork hypersurfaces

In this paper, we calculate the Weiner index, some degree-based topological indices, and some eccentricity-based topological indices of Join graph of Γ(F1 ×F2 ×F3) and Γ(J1 ×J2 ×J3), where F1, F2, F3, J1, J2, J3 are finite fields of order at least two. The vertex set of Γ(F1×F2×F3) (Γ(J1×J2 ×J3)) is Z∗(F1 ×F2 ×F3) (Z∗(J1 ×J2 ×J3)), the set of non-zero zero divisors, and two distinct vertices (x1, x2, x3), (y1, y2, y3) are adjacent if and only if (x1, x2, x3) · (y1, y2, y3) = (0, 0, 0), the additive identity of the ring F1 × F2 × F3 (J1 × J2 × J3). The vertex set of Join graph Γ(F1 × F2 × F3) + Γ(J1 × J2 × J3) is V (Γ(F1 × F2 × F3)) ∪ V (Γ(J1 × J2 × J3)) and edge set is E(Γ(F1 × F2 × F3)) ∪ E(Γ(J1 × J2 × J3)) ∪ {(x1, x2, x3) · (y1, y2, y3) : (x1, x2, x3) ∈ V (Γ(F1 × F2 × F3)), (y1, y2, y3) ∈ V (Γ(J1 × J2 × J3))}.

Let be a finite field of order q, t be a prime and be positive integers. In this article, we find all irreducible divisors of over where and such that are distinct odd primes and c is a positive integer with and Moreover, we construct an a-constacyclic code by using these irreducible divisors.

This thesis focuses on the interplay of extremal combinatorics and finite geometry. Combinatorics is concerned with discrete (and usually finite) objects. Extremal combinatorics studies how large or how small a collection of finite objects can be under certain restrictions. Those objects can be sets, graphs, vectors, etc. These questions are often motivated by problems in information theory and computer science. Another branch of combinatorics is finite geometry over finite fields of order q. Although there is no field of order 1, certain substructures in finite geometry can be interpreted as versions of projective spaces for q = 1. A triangle in a projective plane is a good example of this phenomenon. Following an introductory chapter, Chapter 2 gives an overview of some classical problems and their q-analogues in extremal combinatorics. The most recent results for the q-analogues of Sperner's theorem, the Erdös-Ko-Rado theorem and several versions of Bollobás's theorem are described. For the q-analogue of the Hilton-Milner theorem we give some new results. We also give a new bound for the minimum size of the q-analogue of small maximal cliques. We conclude that sometimes results for a q-analogue can be obtained by using the same technique as in the original problem. In some cases the answer to the problem is even identical to that of the original problem. In other cases techniques used for the q-analogue could be used for improving bounds for the original problem. The third chapter describes the known results for the chromatic number of the Kneser graphs and gives new bounds for the chromatic number of the q-analogue. The vertices of a Kneser graph are subsets (of a fixed size) of a set, whereas two vertices are adjacent if they are disjoint in their subset representation. In 1955 M. Kneser conjectured the chromatic number of those graphs. In 1978 this was proven correct by L. Lóvasz. Two small cases in the projective space q-analogue were solved in 2001 by J. Eisfeld, L. Storme and P. Sziklai and in 2006 by A. Chowdhury, C. Godsil and G. Royle. We describe an asymptotic result for all cases (except for one parameter family, where we give a partial proof) using the bounds from the q-analogue of the Hilton-Milner theorem. Chapters 4, 5 and 6 describe other q-analogues of the Kneser graphs. In Chapter 4 we define a family of Kneser type graphs over pairs of incident points and hyperplanes of a projective space. We describe large maximal cocliques in these graphs and prove that for small dimensions these are the largest possible cocliques. We conjecture that this is the case for all dimensions. In Chapter 5 we extend the Kneser graphs to the case of finite polar spaces instead of finite projective spaces and in some cases give descriptions and chromatic numbers of these graphs. Chapter 6 describes the generalization of Kneser graphs over coset spaces of Chevalley groups (parameterized by q) with respect to parabolic subgroups. This encompasses all previous cases and extends to some interesting new cases. First we describe these graphs when q = 1 and give chromatic numbers for some families. Then we consider the graphs defined over coset spaces of Chevalley groups for general q with respect to parabolic subgroups and study what results of the q = 1 case can be translated to the general case. In this thesis we found different possibilities for the connection between the bounds for the q = 1 case and the case for general q. In some cases the bounds for the general case are identical to the bounds for q = 1. In other cases we obtain the bounds for q = 1 by taking the limit for q¿1 in the general bound. Other cases show a completely different bound in both cases.

Commutative presemifields and semifields

We deal with first-order definability in the substructure ordering \(({\mathscr{D}}; \sqsubseteq )\) of finite directed graphs. In two papers, the author has already investigated the first-order language of the embeddability ordering \(({\mathscr{D}}; \leq )\). The latter has turned out to be quite strong, e.g., it has been shown that, modulo edge-reversing (on the whole graphs), it can express the full second-order language of directed graphs. Now we show that, with finitely many directed graphs added as constants, the first order language of \(({\mathscr{D}}; \sqsubseteq )\) can express that of \(({\mathscr{D}}; \leq )\). The limits of the expressive power of such languages are intimately related to the automorphism groups of the orderings. Previously, analogue investigations have found the concerning automorphism groups to be quite trivial, e.g., the automorphism group of \(({\mathscr{D}}; \leq )\) is isomorphic to \(\mathbb {Z}_{2}\). Here, unprecedentedly, this is not the case. Even though we conjecture that the automorphism group is isomorphic to \((\mathbb {Z}_{2}^{4} \times S_{4})\rtimes _{\alpha } \mathbb {Z}_{2}\), with a particular α in the semidirect product, we only prove it is finite.

On the number of curves of genus 2 over a finite field

It is somnewhat surprising that the systematic evaluation of the heights of rational points on a curve and on its jacobian variety and particularly of their relation to each other should yield any new information. Nonetheless this appears to be the case and the result is described in this article. Although the main theorem is not even a special case of the very fascinating conjecture of Mordell, still it is an estimate that already reveals that rational points on curves of genus at least 2 are much harder to come by than on curves of genus 0 or 1. It is a quantitative limitation on the heights of such points which is well-known to be false in the case of genus 0 or 1. Incidentally, there is a good explanation why an estimate of this type can be obtained so cheaply, whereas Mordell's conjecture itself could not: namely, results obtained by our methods will more or less automatically apply to the analogous funetion case [where the ground field is a function field in one variable over a finite field, rather than an algebraic number field]. And in this case, unless further restrictions are imposed, there are curves of any genus with an infinite number of rational poilnts whose heights increase exactly at the rate which we will find. Let k be an algebraic number field of finite degree over Q. Let C be a non-singular projective curve over k of genus g at least 2. Mordell's conjecture asserts that the set of k-rational points on C is finite. Now suppose

We establish analogs of the three Bieberbach theorems for a lattice \(\Gamma\) in a semidirect product \(\mathsf{K}\rtimes\mathsf{K}\) where \(\mathsf{S}\) is a connected, simply connected solvable Lie group and \(\mathsf{K}\) is a compact subgroup of its automorphism group. We first prove that the action of \(\Gamma\) on \(\mathsf{S}\) is metrically equivalent to an action of \(\Gamma\) on a supersolvable Lie group. The latter is shown to be determined by \(\Gamma\) itself up to an affine diffeomorphism. Then we characterize these lattices algebraically as polycrystallographic groups. Furthermore, we realize any polycrystallographic group \(\Gamma\) as a lattice in a semidirect product \(\mathsf{S}\rtimes\mathsf{F}\) with \(\mathsf{F}\) being a finite group whose order is bounded by a constant only depending on the dimension of \(\mathsf{S}\). This generalization of the first Bieberbach theorem is used to obtain a partial generalization of the third one as well. Finally we show for any torsion free closed subgroup \(\Upsilon \subset \mathsf{K}\rtimes\mathsf{K}\) that the quotient \(\mathsf{S}/\Upsilon\) is the total space of a vector bundle over a compact manifold B, where B is the quotient of a solvable Lie group by a torsion free polycrystallographic group.

The classification of automorphism groups of rational elliptic surfaces with section