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Inflection divisors of linear series on an elliptic curve

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 $E_λ: y^2= x(x-1)(x-λ)$, there is an associated {\it inflectionary curve} in (the projective compactification of) the affine plane in coordinates $x$ and $λ$. 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 $E_λ$ whenever the Legendre parameter $λ$ is real.