Inflection divisors of linear series on an elliptic curve

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.

A separability conjecture for inflection polynomials associated with maximally-real elliptic curves was stated correctly in the first version but not in the second; so we reverted to the first version. Additional small additions/fixes; to appear in the proceedings of the ICM satellite conf in Campinas