On supersingular isogeny graphs of Drinfeld modules

Abstract

We study supersingular isogeny graphs of rank-two Drinfeld modules over $A={\mathbb{F}}_q[T]$. For distinct finite primes $\mathfrak{p}$ and $\mathfrak{q}$ of $A$, we prove that the graph in characteristic $\mathfrak{p}$, with edges given by cyclic $\mathfrak{q}$-isogenies, is connected. The proof combines Gekeler's ideal-class correspondence with strong approximation to realize the graph as a quotient of the Bruhat--Tits tree of the homothety classes of $A_{\mathfrak{q}}$-lattices in $({\mathbb{F}}_q(T){)}_{\mathfrak{q}}^2$. We also introduce the completeness number $E(\mathfrak{p})$, the least integer such that the graph is complete for every prime $\mathfrak{q}\ne \mathfrak{p}$ with $\mathrm{deg}\mathfrak{q}\ge E(\mathfrak{p})$, and derive explicit upper bounds using the Ramanujan--Petersson bound for the eigenvalues of operators associated to Brandt matrices over function fields. In particular, if $d=\mathrm{deg}\mathfrak{p}$, then $E(\mathfrak{p})\le 2d+2$, with sharper bounds depending on $q$ and the parity of $d$. This confirms a conjecture of Micheli and Papikian that the graph becomes complete once $\mathrm{deg}\mathfrak{q}$ is sufficiently large relative to $\mathrm{deg}\mathfrak{p}$. We further obtain parity-dependent lower bounds and prove that $E(\mathfrak{p})\ge d+{\log }_{q}d-K$ for an absolute constant $K>0$ and sufficiently large odd $d$, thereby refuting the previously suggested bound $E(\mathfrak{p})\le d+1$. Finally, we prove a criterion yielding an algorithm to compute $E(\mathfrak{p})$.

arXiv comment19 pages