Explicit moduli spaces for congruences of elliptic curves

TL;DR

This paper constructs explicit models of modular surfaces over Q that parametrize pairs of N-congruent elliptic curves, providing detailed geometric and arithmetic properties such as rank and Picard number.

Contribution

It provides explicit birational models for these modular surfaces in all cases where they are elliptic surfaces, including their Mordell-Weil rank and Picard number.

Findings

Explicit birational models over Q for modular surfaces

Determination of Mordell-Weil rank for each surface

Calculation of geometric Picard number

Abstract

We determine explicit birational models over Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell-Weil lattice and the geometric Picard number.