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 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 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 real inflection points of linear series on E λ whenever the Legendre parameter λ is real.

Arithmetic inflection of superelliptic curves

“…The geometry of superelliptic Legendre pencils is closely linked to algebraic differential equations and hypergeometric series; see, e.g., [12,17]. The conjectural number of singularities (3) of Weierstrass inflectionary curves appears in blue as it is not stated explicitly in our earlier papers [3,6,7,5]. However iterating the characteristic recursion for atomic inflection polynomials leads to the expectation (formalized in Conjecture 4.17 below) that the corresponding inflectionary curves C m with m ≥ 3 are always singular exactly in the points q j = [ζ −j : −3ζ j : 1], j = 0, 1, 2 in the weighted projective plane P(1, 2, 1) 6 , where ζ is a cube root of unity.…”

“…In the papers [3,5,6,7], we studied F -rational inflectionary loci for certain linear series on hyperelliptic curves X defined over F . Whenever char(F ) = 2 and a hyperelliptic curve X has an F -rational ramification point ∞ X , X admits an affine model y 2 = f (x) in ambient coordinates x and y with respect to which the complete series | ∞ X | has a distinguished basis of monomials in x and y.…”

“…Examples 3.4 and 3.5 lead to interesting identities involving Vandermonde determinants; see, e.g., equation (11). Indeed, every entry of M (p) := W p * (B) is defined purely combinatorially by equation (7). Now let M (p) denote the matrix obtained from M (p) by systematically replacing every monomial Π i (D i y x) ci in Hasse derivatives of x by the corresponding monomial Π i t ci i in formal variables t i .…”

“…In [5,6,7], we studied F -rationality phenomena for inflectionary curves C m defined by atomic inflection polynomials P m built out of one-parameter Legendre and Weierstrass pencils of elliptic curves, with a focus on those cases in which F = R or F = F p for an odd prime p. In this setting, C m is naturally a singular plane curve defined over Z, or else its reduction modulo p. Moreover, the birational geometry of inflectionary curves C m varies depending upon whether the underlying pencil of elliptic curves is of Legendre or Weierstrass type. In particular, the inflectionary curves C m , 2 ≤ m ≤ 5 derived from the Legendre pencil have rational desingularizations, whereas the Weierstrass inflectionary curve C 2 is elliptic, with complex multiplication over Q(…”

Arithmetic inflection formulae for linear series on hyperelliptic curves

“…According to the analogy between torsion and inflection introduced in Section 1.5, we may think of these inflectionary curves as generalizations of modular curves. Whenever κ is a real parameter, so that the corresponding elliptic curve $$E_{\kappa }$$ has a real locus $$E_{\kappa }(\mathbb {R})$$ with two connected components, [10, Conj. 3.1] predicts that the corresponding inflection polynomial is separable, that is, has only simple roots.…”