In this paper we give a complete characterization of the intersections between Norm-Trace curve over $\mathbb{F}_{q^3}$ and curves form $y=ax^3+bx^2+cx+d$, generalizing previous result by Bonini Sala, providing more detailed information about weight spectrum one-point AG codes arising from such curve. We also derive, with explicit computations, some general bounds for number rational points on ...

(2) hn divides hm whenever n divides m. They have attracted number theoretical and combinatorial interest as some of the simplest nonlinear recurrence sequences (see [3] for references), but for us their interest lives in the underlying geometry: Ward demonstrates that an elliptic divisibility sequence arises from any choice of elliptic curve over Q and rational point on that curve. Theorem 1 (...

1.1. Given an abelian variety A over a function field K = k(C) with C an absolutely irreducible, smooth, proper curve over a field k, it is natural to ask about the behavior of the Mordell-Weil group of A in the layers of a tower of fields over K. The simplest case, which is already very interesting, is when A is an elliptic curve, K = k(t) is a rational function field, and one considers the to...

Let k = Fq be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C2 × C2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an ArtinSchreier cover of the three elliptic curves obtained as the quotient of C by the nontrivial involutions of H , and the Jacobian of C is k-isogenous to the prod...

Recently Perlmutter et al. (1999) suggested a positive value of Einstein’s cosmological constant Λ on the basis of luminosity distances from type-Ia supernovae (the “SN-method”). However, Λ world models had earlier been proposed by Hoell & Priester (1991) and Liebscher et al. (1992a,b) on the basis of quasar absorption-line data (the “Q-method”). Employing more general repulsive fluids (“dark e...

Abstract In Beelen and Montanucci (Finite Fields Appl 52:10–29, 2018) Giulietti Korchmáros (Math Ann 343:229–245, 2009), Weierstrass semigroups at points of the Giulietti–Korchmáros curve $${\mathcal {X}}$$ X were investigated sets minimal generators determined for all in {X}}(\mathbb {F}_{q^2})$$ ...

Let P be a polygonal curve in R of length n, and S be a point set of size k. We consider the problem of finding a polygonal curve Q on S such that all points in S are visited and the Fréchet distance between Q and P is at most a given ε. We show that this problem is NP-complete, regardless of whether or not the points of S are allowed to be visited more than once.

Let k = Fq be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C2 × C2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an ArtinSchreier cover of the three elliptic curves obtained as the quotient of C by the nontrivial involutions of H , and the Jacobian of C is k-isogenous to the prod...

Around 1940, Lind [Lin] and (independently, but shortly later) Reichardt [Re] discovered that some genus 1 curves over Q, such as 2y = 1− 17x, violate the Hasse principle; i.e., there exist x, y ∈ R satisfying the equation, and for each prime p there exist x, y ∈ Qp satisfying the equation, but there do not exist x, y ∈ Q satisfying the equation. In fact, even the projective nonsingular model h...

act on the upper half-plane by linear fractional transformations. We may form the quotient Y0(N) = Γ0(N)\H, a Riemann surface with finitely many cusps. Compactifying gives a curve X0(N) which is in fact defined over Q; the map z → (j(z), j(Nz)) ∈ AC realizes Y0(N) as a (highly singular) plane curve with Q-coefficients. Over a general field k of characteristic zero, the k-points of the curve X0(...