An Explicit Manin-Dem'janenko theorem in elliptic curves

TL;DR

This paper provides explicit bounds for rational points on certain high-genus curves embedded in products of elliptic curves, advancing the Manin-Dem'janenko method with concrete quantitative results.

Contribution

It offers an explicit version of the Manin-Dem'janenko theorem for curves in products of elliptic curves, including bounds on rational points and intersections with algebraic subgroups.

Findings

Explicit upper bounds for rational points on specific curves.

Quantitative bounds for intersections with algebraic subgroups.

Enhanced Manin-Dem'janenko method with explicit constants.

Abstract

Let C be a curve of genus at least 2 imbedded in a product of elliptic curves. We give an explicit upper bound for the points in the intersection of C with the union of all algebraic subgroups of a certain codimension. As a corollary we obtain an explicit upper bound for the height of the rational points for some families of curves and an explicit version of the Manin-Dem'janenko method in products of elliptic curves