Genus periods, genus points and congruent number problem

The Congruent Number Problem

We discuss a famous problem about right triangles with rational side lengths. This elementarysounding problem is still not completely solved; the last remaining step involves the Birch and Swinnerton-Dyer conjecture, which is one of the most important open problems in number theory (right up there with the Riemann hypothesis). 6.

The Maximum Number of Points on a Curve of Genus

Our aim in this paper is to prove that a smooth geometrically irreducible curve C of genus 4 over the finite field F8 may have at most 25 F8-points. Our strategy is as follows: if C has more than 18 F8-points, then C may not be hyperelliptic, and so the canonical divisor of C yields an embedding of C into P3F8 . The image of C under this embedding is a degree 6 curve which is precisely the inte...

Integral points on congruent number curves

We provide a precise description of the integer points on elliptic curves of the shape y2 = x3 − N2x, where N = 2apb for prime p. By way of example, if p ≡ ±3 (mod 8) and p > 29, we show that all such points necessarily have y = 0. Our proofs rely upon lower bounds for linear forms in logarithms, a variety of old and new results on quartic and other Diophantine equations, and a large amount of ...

The Genus Crossing Number

Pach and Tóth [6] introduced a new version of the crossing number parameter, called the degenerate crossing number, by considering proper drawings of a graph in the plane and counting multiple crossing of edges through the same point as a single crossing when all pairwise crossings of edges at that point are transversal. We propose a related parameter, called the genus crossing number, where ed...

Genus, Treewidth, and Local Crossing Number