# A variant of a theorem by Ailon-Rudnick for elliptic curves

## TL;DR

This paper proves a new variant of a theorem related to elliptic curves over number fields, establishing conditions under which certain algebraic points imply isogenies or scalar multiples, thereby addressing a conjecture by Silverman.

## Contribution

It introduces a novel variant of the Ailon-Rudnick theorem for elliptic curves, providing new criteria linking algebraic points to isogenies and scalar multiples, and confirms a conjecture by Silverman.

## Key findings

- If infinitely many algebraic points satisfy certain conditions, then an isogeny exists between the elliptic surfaces.
- Alternatively, the points are scalar multiples of each other on the elliptic surfaces.
- The result confirms a specific conjecture by Silverman.

## Abstract

Given a smooth projective curve C defined over a number field and given two elliptic surfaces E_1/C and E_2/C along with sections P_i and Q_i of E_i (for i = 1,2), we prove that if there exist infinitely many algebraic points t on C such that for some integers m_{1,t} and m_{2,t}, we have that [m_{i,t}](P_i)_t = (Q_i)_t on E_i (for i = 1,2), then at least one of the following conclusions must hold: either (i) there exists an isogeny f between E_1 and E_2 and also there exists a nontrivial endomorphism g of E_2 such that f(P_1) = g(P_2); or (ii) Q_i is a multiple of P_i for some i = 1,2. A special case of our result answers a conjecture made by Silverman.