Abstract. We discuss arithmetic in the Jacobian of a hyperelliptic curve C of genus g. The traditional approach is to fix a point P∞ ∈ C and represent divisor classes in the form E − d(P∞) where E is effective and 0 ≤ d ≤ g. We propose a different representation which is balanced at infinity. The resulting arithmetic is more efficient than previous approaches when there are 2 points at infinity.
It is well known that a Severi-Brauer surface has a rational point if and only if it is isomorphic to the projective plane. Given a Severi-Brauer surface, we study the problem to decide whether such an isomorphism to the projective plane, or such a rational point, does exist; and to construct such an isomorphism or such a point in the affirmative case. We give an algorithm using Lie algebra techniques. The algorithm has been implemented in Magma.
A fibration of R n by oriented copies of R p is called skew if no two fibers intersect nor contain parallel directions. Conditions on p and n for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of R 3 by skew oriented lines was given by Salvai, in analogue with the classification of oriented great circle fibrations of S 3 by Gluck and Warner. We show that Salvai's classification has a topological variation which generalizes to characterize all continuous fibrations of R n by skew oriented copies of R p . We show that the space of fibrations of R 3 by skew oriented lines deformation retracts to the subspace of Hopf fibrations, and therefore has the homotopy type of a pair of disjoint copies of S 2 . We discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of C
A smooth fibration of R3 by oriented lines is given by a smooth unit vector field V on R3, for which all of the integral curves are oriented lines. Such a fibration is called skew if no two fibers are parallel, and it is called nondegenerate if ∇V vanishes only in the direction of V. Nondegeneracy is a form of local skewness, though in fact any nondegenerate fibration is globally skew. Nondegenerate and skew fibrations have each been recently studied, from both geometric and topological perspectives, in part due to their close relationship with great circle fibrations of S3.
Any fibration of R3 by oriented lines induces a plane field on R3, obtained by taking the orthogonal plane to the unique line through each point. We show that the plane field induced by any nondegenerate fibration is a tight contact structure. For contactness we require a new characterization of nondegenerate fibrations, whereas the proof of tightness employs a recent result of Etnyre, Komendarczyk and Massot on tightness in contact metric 3‐manifolds.
We conclude with some examples which highlight relationships among great circle fibrations, nondegenerate fibrations, skew fibrations and the contact structures associated to fibrations.
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