Elliptic Curves, Riordan arrays and Lattice Paths

TL;DR

This paper establishes a connection between elliptic curves, lattice paths, and Riordan arrays, showing that their associated Somos 4 sequences are fundamentally linked through elliptic divisibility sequences and Hankel transforms.

Contribution

It introduces a novel association between elliptic curves and lattice paths via Riordan arrays, revealing a shared Somos 4 sequence structure.

Findings

Elliptic curves can be linked to lattice paths through Riordan arrays.

Both curves and paths share the same Somos 4 sequence structure.

The connection involves elliptic divisibility sequences and Hankel transforms.

Abstract

In this note, we show that to each elliptic curve of the form $$y^2-axy-y=x^3-bx^2-cx,$$ we can associate a family of lattice paths whose step set is determined by the parameters of the elliptic curve. The enumeration of these lattice paths is by means of an associated Riordan array. The curves and the paths have associated Somos $4$ sequences which are essentially the same. For the curves the link to Somos $4$ sequences is a classical result, via the elliptic divisibility sequence. For the paths the link is via a Hankel transform.