On supersingular isogeny graphs of Drinfeld modules
Abstract
We study supersingular isogeny graphs of rank-two Drinfeld modules over A=𝔽q[T]. For distinct finite primes p and q of A, we prove that the graph in characteristic p, with edges given by cyclic 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 q-lattices in (𝔽q(T)) q2. We also introduce the completeness number E(p), the least integer such that the graph is complete for every prime q≠p with degq≥ E(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=degp, then E(p)≤ 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 degq is sufficiently large relative to degp. We further obtain parity-dependent lower bounds and prove that E(p)≥ d+logq d−K for an absolute constant K>0 and sufficiently large odd d, thereby refuting the previously suggested bound E(p)≤ d+1. Finally, we prove a criterion yielding an algorithm to compute E(p).
Related
- The Pila-Zannier strategy for Drinfeld modules and Drinfeld modular curves
- Coset representatives corresponding to Yetter-Drinfeld modules of modular group and continued fraction
- Cyclicity of Drinfeld modules modulo primes in arithmetic progressions
- Skew cyclic codes through Drinfeld modules
- Rank-Two Drinfeld Modules over Elliptic Coordinate Rings
BibTeX
@misc{veselinov2026supersingular,
title = {{On supersingular isogeny graphs of Drinfeld modules}},
author = {Nikola Veselinov},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29812},
eprint = {2608.29812},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29812},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On supersingular isogeny graphs of Drinfeld modules
AU - Nikola Veselinov
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29812
UR - https://arxiv.org/abs/2608.29812
AB - We study supersingular isogeny graphs of rank-two Drinfeld modules over A=𝔽_q[T]. For distinct finite primes p and q of A, we prove that the graph in characteristic p, with edges given by cyclic 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_ q-lattices in (𝔽_q(T))_ q^2. We also introduce the completeness number E(p), the least integer such that the graph is complete for every prime q≠p with degq≥ E(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=degp, then E(p)≤ 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 degq is sufficiently large relative to degp. We further obtain parity-dependent lower bounds and prove that E(p)≥ d+log_q d−K for an absolute constant K>0 and sufficiently large odd d, thereby refuting the previously suggested bound E(p)≤ d+1. Finally, we prove a criterion yielding an algorithm to compute E(p).
ER -