Geometric meaning of Selling-reduction transformation data

Determine whether there exist subfamilies of the split-Jacobian locus that restrict the possible sequences of Selling-reduction transformations and whether the resulting sequence of integers has a geometric meaning.

Background

The paper describes the split-Jacobian locus $\mathcal{Q}$ using parameters associated with two tropical elliptic curves and an isomorphism between their $d$-torsion subgroups. To reduce the associated quadratic form to the standard cone, Selling’s algorithm produces a sequence $(\alpha_1,\beta_1,\ldots,\alpha_n,\beta_n)$ of positive integers recording elementary transformations.

Explicit descriptions are obtained for certain subfamilies, notably those corresponding to $k=1$ and $k=d-1$, because the possible reduction data are then constrained. The paper leaves unresolved whether other subfamilies similarly restrict the reduction sequences and whether those sequences encode intrinsic geometric information rather than merely auxiliary algebraic reduction data.