Elliptic subcovers of a curve of genus 2. II: The refined Humbert invariant
Let \(C/K\) be a curve of genus 2 over an arbitrary field \(K\). The first part of this two-part paper [\textit{E. Kani}, Ann. Math. Qué. 43, No. 2, 281--303 (2019; Zbl 1435.14029)] established a bijection between the set of equivalence classes of the elliptic subcovers of \(C/K\) and the set of certain primitive representations of an intrinsic quadratic form \(q_C\) called the \textit{refined Humbert invariant}. This second part explains how to compute the refined Humbert invariant explicitly from an ``\(N\)-presentation'' of the Jacobian variety \(J\) of \(C\) as \(J\cong (E\times E')/\mathrm{Graph}(-\psi)\), where \((E,E',\psi)\) is a certain triple with \(E, E'\) isogenious elliptic curves and \(\psi:E[N]\overset{\sim}{\rightarrow} E'[N]\) (\(1<N\in\mathbb{Z}\)). The domain lattice of \(q_C\) is the saturation \(NS'(J)\) of the Neron-Severi group \(NS(J)\) in \(NS(J_{\bar{K}})\) modulo the theta line whose rank is \(1+\mathrm{rank}_\mathbb{Z}(\mathrm{Hom}_K(E,E'))\). The main ingredient (Theorem 25) explicitly computes the coefficients \(a,b,c\) of a binary quadratic form \(q_C(X\bar D_1+Y\bar D_2)=aX^2+bXY+cY^2\) for the images \(\bar D_1\), \(\bar D_2\) of certain divisors \(D_1,D_2\in NS'(J)\) given with relevant numerical invariants. In the final paragraph of the paper is illustrated a very impressive and representative example for a Legendre curve \(C_K:y^2=x(x^2-3)(x^2-\frac13)\) over \(K\): If \(K\not\supset\mathbb{Q}(\sqrt{-1})\) then \(q_{C_K}\sim 4X^2\); if \(\mathbb{Q}(\sqrt{-1})\subset K \not\supset\mathbb{Q}(\sqrt{-3})\) then \(q_{C_K}\sim 4X^2+4Y^2\); and if \(\mathbb{Q}(\sqrt{-1},\sqrt{-3})\subset K\) then \(q_{C_K}\sim 4X^2+4Y^2+4YZ+4Z^2\).
scientific article; zbMATH DE number 6929040