The rational points close to a curve II

Staying Close to a Curve

Given a point set S and a polygonal curve P in R, we study the problem of finding a polygonal curve through S, which has a minimum Fréchet distance to P . We present an efficient algorithm to solve the decision version of this problem in O(nk) time, where n and k represent the sizes of P and S, respectively. A curve minimizing the Fréchet distance can be computed in O(nk log(nk)) time. As a by-...

Distribution of Rational Points: A Survey

Independence of Rational Points on Twists of a given Curve

In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of K-rational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form 2r + c, where r is the rank of J ′(K) a...

How Many Rational Points Does a Random Curve Have?

A large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties. For an elliptic curve over Q, the set of rational points forms a finitely generated abelian group. The ranks of these groups, when ranging over all elliptic curves, are conjectured to be evenly distributed between rank 0 and rank 1, with higher ranks being negligible. We will de...

The Ordering of Points on a Curve. Part II

(1) Let P, Q be subsets of E2 T, p1, p2, q1 be points of E2 T, f be a map from I into (E2 T) P, and s1 be a real number. Suppose that P is an arc from p1 to p2 and q1 ∈ P and q1 ∈ Q and f (s1) = q1 and f is a homeomorphism and f (0) = p1 and f (1) = p2 and 0 ≤ s1 and s1 ≤ 1 and for every real number t such that 0 ≤ t and t < s1 holds f (t) / ∈ Q. Let g be a map from I into (E2 T) P and s2 be a ...