We construct curves with many points over finite fields by using the class group.

Let p be a prime and Fq be a finite field with q = p m elements and let ¯ Fq be an algebraic closure of Fq. In this paper we present a method for constructing curves over finite fields with many points which are Kummer covers of P 1, or of other suitable base curves. For this we look at rational

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this paper we investigate some plane curves with many points over Q, finite fields and cyclotomic fields

dx.doi.org/10.2140/obs.2013.1.317 msp Computing equations of curves with many points

Any set of labeled examples consistent with some hidden orthogonal rectan-gle can be “compressed ” to at most four examples: An upmost, a leftmost, a rightmost and a bottommost positive example. These four examples represent an orthogonal rectangle (the smallest such rectangle that contains them) that is consistent with all examples. Note that the VC dimension of orthogonal rectangles is four and this is exactly the number of examples needed to represent the consistent orthogonal rectangle. A compression scheme of size k for a concept class C picks from any set of examples consistent with some concept in C a subset of up to k examples and this subset represents (via a mapping that that is specific to the class C) a hypothesis consistent with the whole original set of examples. Conjecture: Any concept class of VC dimension d has a compression scheme of size d. What evidence do we have that this conjecture might be true? Call a concept class of VC dimension d maximum if for every subset of m instances, the concept class induces exactly ∑d i=1

Abstract. We explain how to compute the equations of the abelian coverings of any curve defined over a finite field. Then we describe an algorithm which computes curves with many rational points with respect to their genus. The implementation of the algorithm provides 7 new records over F2.

Let A be a given compact subset of the euclidean space. We consider the problem of finding a compact connected set S of minimal 1-dimensional Hausdorff measure, among all compact connected sets containing A. We prove that when A is a finite set any minimizer is a finite tree with straight edges, thus recovery the classical Steiner Problem. Analogously, in the case when A is countable, we prove that every minimizer is a (possibly) countable union of straight segments.

Let S be a set of points in the plane in general position. A triangulation of S will be called even if all the points of S have an even degree. We show how to construct a triangulation of S containing at least ⌊ 2n 3 ⌋−3 points with even degree; this improves slightly the bound of ⌈ 2(n−1) 3 ⌉ − 6

Let G be a finite set of points in the plane. A line M is a (k, k)-line, if M is determined by G, and there are at least k points of G in each of the two open half-planes bounded by M. Let f(k, k) denote the maximum size of a set G in the plane, which is not contained in a line and does