Stabilization of isogeny spaces between supersingular Drinfeld modules

Abstract

Let $\mathfrak{p}$ be a prime of degree $d$ in $A={\mathbb{F}}_q[T]$ and let $\ddot{I}•,\ddot{I}ˆ$ be supersingular Drinfeld modules of rank $r\ge 2$ in $A$-characteristic $\mathfrak{p}$. We study the ${\mathbb{F}}_q$-dimension of the space $M_s(\ddot{I}•,\ddot{I}ˆ)=\{u\in \mathrm{Hom}(\ddot{I}•,\ddot{I}ˆ):{\mathrm{deg}}_{\ddot{I}}„u\le s\}$ as a function of $s$. By analyzing $\mathrm{Hom}(\ddot{I}•,\ddot{I}ˆ)$ as a normed $A$-lattice in the local division algebra at $\infty$ via its successive minima, we obtain an exact closed-form expression for ${\dim }_{{\mathbb{F}}_q}{M}_s(\ddot{I}•,\ddot{I}ˆ)$ valid for every $s\ge 0$, together with structural constraints on the successive-minima multiset which imply the stabilization formula ${\dim }_{{\mathbb{F}}_q}{M}_s(\ddot{I}•,\ddot{I}ˆ)=r(s+1)-\frac{r(r-1)(d-1)}{2}$ for all $s\ge r^2(r-1)(d-1)/2$. We conjecture that the optimal threshold is $s\ge (r-1)(d-1)-1$, 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.

v2: substantially revised. Main result generalized to arbitrary rank via successive minima in the local division algebra; added geometric reinterpretation (cohomology vanishing on P^1). Code constructions removed, deferred to forthcoming work. Title changed