On Symmetries of Elliptic Nets and Valuations of Net Polynomials

TL;DR

This paper explores the symmetry properties and zero sets of elliptic nets, extending classical theorems on elliptic divisibility sequences and valuations of division polynomials to a broader context.

Contribution

It proves that zeros of elliptic nets form an Abelian group and generalizes key theorems of Ayad and Ward to elliptic nets.

Findings

Zeros of elliptic nets form an Abelian group

Generalization of Ayad's valuation theorem to net polynomials

Extension of Ward's symmetry theorem to elliptic nets

Abstract

Under certain conditions, we prove that the set of zeros of an elliptic net forms an Abelian group. We present two applications of this fact. Firstly we give a generalization of a theorem of Ayad on valuations of division polynomials in the context of net polynomials. Secondly we generalize a theorem of Ward on symmetry of elliptic divisibility sequences to the case of elliptic nets.