# Inflection divisors of linear series on an elliptic curve

## TL;DR

This paper explores special divisors on elliptic curves that mark inflection points of certain linear series, leading to explicit conjectures about real inflection points based on associated inflectionary curves.

## Contribution

It introduces a class of inflection divisors on elliptic curves derived from pullbacks of line bundles on , and relates them to inflectionary curves that inform conjectures on real inflection points.

## Key findings

- Definition of inflection divisors on elliptic curves
- Construction of inflectionary curves in the affine plane
- Formulation of an explicit conjecture for real inflection points

## Abstract

In this largely-expository note, we describe a class of divisors on elliptic curves that index the inflection points of linear series arising (as subspaces of holomorphic sections) from line bundles on $\mathbb{P}^1$ via pullback along the canonical 2-to-1 projection. Associated to each inflection divisor on an elliptic curve $E_{\lambda}: y^2= x(x-1)(x-\lambda)$, there is an associated {\it inflectionary curve} in (the projective compactification of) the affine plane in coordinates $x$ and $\lambda$. These inflectionary curves have remarkable features; among other things, they lead directly to an explicit conjecture for the number of {\it real} inflection points of linear series on $E_{\lambda}$ whenever the Legendre parameter $\lambda$ is real.