AI 中文总结
本文研究A=𝔽_q[T]上秩2的Drinfeld模的超奇异同构图，证明其连通性，引入完备性数并推导其上、下界，证实相关猜想并反驳旧界，还给出计算该数的算法准则。
AI 中文摘要
我们研究A=𝔽_q[T]上秩为2的Drinfeld模的超奇异同构图。对于A中不同的有限素理想𝔭和𝔮，我们证明以循环𝔮-同构为边的特征𝔭下的该图是连通的。该证明结合了Gekeler的理想类对应与强逼近，将该图实现为(𝔽_q(T))_𝔮²中A_𝔮-格的同伦类的Bruhat–Tits树的商。我们还引入完备性数E(𝔭)，即使得对所有满足deg𝔮≥E(𝔭)且𝔮≠𝔭的素理想𝔮，该图是完备的最小整数，并利用函数域上与Brandt矩阵相关算子的Ramanujan–Petersson界推导显式上界。特别地，若d=deg𝔭，则E(𝔭)≤2d+2，且存在依赖q与d奇偶性的更紧界。这证实了Micheli和Papikian的猜想：当deg𝔮相对于deg𝔭足够大时，该图会变为完备。我们进一步得到依赖奇偶性的下界，证明当d足够大且为奇数时，E(𝔭)≥d+log_q d-K（K为正绝对常数），从而反驳了此前提出的E(𝔭)≤d+1的界。最后，我们证明一个可用于计算E(𝔭)的算法准则。
英文摘要
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 $E(\mathfrak p)$.
Comments19 pages
