Stabilization of isogeny spaces between supersingular Drinfeld modules

Let $\mathfrak{p}$ be a prime of degree $d$ in $A = \mathbb{F}_q[T]$ and let $\phi, \psi$ be supersingular Drinfeld modules of rank $r \geq 2$ in $A$-characteristic $\mathfrak{p}$.

We study the $\mathbb{F}_q$-dimension of the space $M_s(\phi, \psi) = \{u \in \mathrm{Hom}(\phi, \psi) : \mathrm{deg}_\tau u \leq s\}$ as a function of $s$.

By analyzing $\mathrm{Hom}(\phi, \psi)$ as a normed $A$-lattice in the local division algebra at $\infty$ via its successive minima, we obtain an exact closed-form expression for $\dim_{\mathbb{F}_q} M_s(\phi, \psi)$ valid for every $s \geq 0$, together with structural constraints on the successive-minima multiset which imply the stabilization formula $\dim_{\mathbb{F}_q} M_s(\phi, \psi) = r(s+1) - \frac{r(r-1)(d-1)}{2}$ for all $s \geq r^2(r-1)(d-1)/2$.

We conjecture that the optimal threshold is $s \geq (r-1)(d-1) - 1$, and prove this sharp form for $r = 2$ by independent automorphic methods, using the decomposition of a Brandt-type theta series on the Bruhat-Tits tree of $\mathrm{PGL}_2(F_\infty)$ into Eisenstein and cuspidal parts together with the polynomiality of the cuspidal $L$-function.

We also recast our results in Mornev's geometric framework, in which the conjecture becomes a cohomology-vanishing statement for a family of vector bundles on $\mathbb{P}^1$, and illustrate the theory with explicit examples in which all successive-minima multisets permitted by our constraints are realized.