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 $\mathfrak q\neq\mathfrak p$ with $\operatorname{deg}\mathfrak q\geq E(\mathfrak p)$, 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 $d=\operatorname{deg}\mathfrak p$, then $E(\mathfrak p)\leq 2d+2$, 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 $\operatorname{deg}\mathfrak q$ is sufficiently large relative to $\operatorname{deg}\mathfrak p$. We further obtain parity-dependent lower bounds and prove that $E(\mathfrak p)\geq d+\log_q d-K$ for an absolute constant $K>0$ and sufficiently large odd $d$, thereby refuting the previously suggested bound $E(\mathfrak p)\leq d+1$. Finally, we prove a criterion yielding an algorithm to compute $E(\mathfrak p)$.