Searcharxiv⌕ Search

arXiv subjects

Robin de Jong

Publications and source records attributed to Robin de Jong.

41 records · Page 3Linked to original sources
arXiv2009

Covers of surfaces with fixed branch locus

Given a connected smooth projective surface X over the complex numbers, together with a simple normal crossings divisor D on it, we study finite normal covers Y of X that are unramified outside D. Given moreover a fibration of X onto a curve C, we prove that the `height' of Y over C is bounded linearly in terms of the degree of Y over X. We indicate how an arithmetic analogue of this result, if true, can be auxiliary in proving the existence of a polynomial time algorithm that computes the mod-l Galois representations associated to a given smooth projective geometrically connected surface over the rational numbers. A precise conjecture is formulated.

math.AG↗
arXiv2007

Theta functions on the theta divisor

We show that the gradient and the hessian of the Riemann theta function in dimension n can be combined to give a theta function of order n+1 and modular weight (n+5)/2 defined on the theta divisor. It can be seen that the zero locus of this theta function essentially gives the ramification locus of the Gauss map. For jacobians this leads to a description in terms of theta functions and their derivatives of the Weierstrass point locus on the associated Riemann surface.

math.AG↗
arXiv2004

On the Arakelov theory of elliptic curves

This note contains an elementary discussion of the Arakelov intersection theory of elliptic curves. The main new results are a projection formula for elliptic arithmetic surfaces and a formula for the "energy" of an isogeny between Riemann surfaces of genus 1. The latter formula provides an answer to a question originally posed by Szpiro.

math.NT↗
arXiv2004

Arakelov invariants of Riemann surfaces

We derive explicit formulas for the Arakelov-Green function and the Faltings delta-invariant of a Riemann surface. A numerical example illustrates how these formulas can be used to calculate Arakelov invariants of curves.

math.NT↗