Sharp stabilization threshold for bounded-degree morphism spaces

Determine whether, for supersingular Drinfeld modules \(\phi\) and \(\psi\) of rank \(r\ge 2\) over an algebraic closure of a finite-characteristic field with characteristic prime of degree \(d\), the dimension formula \(\dim_{\mathbb F_q} M_s(\phi,\psi)=r(s+1)-r(r-1)(d-1)/2\) holds for every integer \(s\ge (r-1)(d-1)-1\).

Background

The paper uses the spaces $M_s(\phi,\psi)$ of geometric morphisms between supersingular Drinfeld modules whose twisted-polynomial degree is at most $s$. A stabilization theorem cited from Micheli and Papikian establishes the dimension formula $r(s+1)-r(r-1)(d-1)/2$ only in the range $s\ge r^2(r-1)(d-1)/2$.

The sharper threshold $s\ge (r-1)(d-1)-1$ would substantially enlarge the range in which the dimension formula is available. The present paper relies on the established, more restrictive stabilization range to derive its rank-metric and sum-rank code parameters; it does not resolve the sharper-threshold conjecture. The paper notes that the conjectured range is known when $r=2$.