Galois points for a normal hypersurface

Galois Points for a Normal Hypersurface

We study Galois points for a hypersurface X with dimSing(X) ≤ dimX − 2. The purpose of this article is to determine the set ∆(X) of Galois points in characteristic zero: Indeed, we give a sharp upper bound of the number of Galois points in terms of dimX and dimSing(X) if ∆(X) is a finite set, and prove that X is a cone if ∆(X) is infinite. To achieve our purpose, we need a certain hyperplane se...

Integral Points of Small Height Outside of a Hypersurface

Let F be a non-zero polynomial with integer coefficients in N variables of degree M . We prove the existence of an integral point of small height at which F does not vanish. Our basic bound depends on N and M only. We separately investigate the case when F is decomposable into a product of linear forms, and provide a more sophisticated bound. We also relate this problem to a certain extension o...

Galois Theory and Torsion Points on Curves

We begin with a brief history of the problem of determining the set of points of a curve that map to torsion points of the curve’s Jacobian. Let K be a number field, and suppose that X/K is an algebraic curve of genus g ≥ 2. Assume, furthermore, that X is embedded in its Jacobian variety J via a K-rational Albanese map i; thus there is a K-rational divisor D of degree one on X such that i = iD ...

Explicit Uniform Estimation of Rational Points Ii. Hypersurface Coverings

Galois theory and the normal basis theorem