We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field ¥q2 whose number of ¥q2 -rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are ¥q2 -isomorphic to y q + y = x for some m £ Z + . MIRAMARE TRIESTE March 1996 1 E-mail: [email protected] E-mail: f...

In practice, laterally loaded piles are most often analysed using a ‘beam-on-non-linear-Winklerfoundation’ approach, whereby the soil–structure interaction is modelled by means of p–y curves. Although well-calibrated p–y curves exist for non-liquefied soils (e.g. soft clay and sand), the profession still lacks reliable p–y curves for liquefied soils. In fact, the latter should be consistent wit...

In this paper we investigate the mapping properties in Lebesgue-type spaces of certain generalized Radon transforms defined by integration over curves. Let X and Y be open subsets of R, d ≥ 2, and let Z be a smooth submanifold of X×Y ⊂ R2d of dimension d+1. Assume that the projections π1 : Z → X and π2 : Z → Y are submersions at each point of Z. For each y ∈ Y , let γy = {x ∈ X : (x, y) ∈ Z} = ...

Let X → Y be a Galois covering of curves, where the genus of X is ≥ 2 and the genus of Y is ≤ 2. We prove that under certain hypotheses, X has an unramified cover that dominates a hyperelliptic curve; our results apply, for instance, to all tamely superelliptic curves. Combining this with a theorem of Bogomolov and Tschinkel shows that X has an unramified cover that dominates y = x − 1, if char...

‎By the Mordell‎- ‎Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎This paper studies the rank of the family Epq:y2=x3-pqx of elliptic curves‎, ‎where p and q are distinct primes‎. ‎We give infinite families of elliptic curves of the form y2=x3-pqx with rank two‎, ‎three and four‎, ‎assuming a conjecture of Schinzel ...

By the Mordell-Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎There is no known algorithm for finding the rank of this group‎. ‎This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

This paper proves that the Montgomery form elliptic curves that are cheaply transformable into short Weierstrass form by a simple change of variables (x, y) 7→ (x+α, y) (instead of a more general affine change of variables) are precisely the curves with B = 1. The points of order 2 and 4 on these curves are described, and it is observed that the x-coordinates of these points are consecutive fie...

Recently two kinds of Huff curves were introduced as elliptic curves models and their arithmetic was studied. It was also shown that they are suitable for cryptographic use such as Montgomery curves or Koblitz curves (in Weierstrass form) and Edwards curves. In this work, we introduce the new generalized Huff curves ax(y − c) = by(x−d) with abcd(ac−bd) 6= 0, which contains the generalized Huff’...

In this paper, the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed. Unlike most of the previous related works, each reducible curve of the pencil is here counted with its appropriate multiplicity. It is proved that this number of reducible curves, counted with multiplicity, is bounded by d − 1 where d is the degree of the pencil. Then, a sha...

Let X be a continuum, that is a compact, connected, nonempty metric space. The span of X is the least upper bound of the set of real numbers r which satisfy the following conditions: there exists a continuum, C, contained in X × X such that d(x, y) is larger than or equal to r for all (x, y) in C and p1(C) = p2(C), where p1, p2 are the usual projection maps. The following question has been aske...