The Richelot isogeny graph and paramodular forms of weight 3: Atkin-Lehner signs and certified spectra
Abstract
The Richelot (2,2)-isogeny graph Gp of superspecial abelian surfaces in characteristic p is the genus-2 analogue of the supersingular isogeny graph: its weighted adjacency matrix is a Brandt matrix for a compact inner form of GSp4, and it commutes with an Atkin-Lehner involution R(π). Refining Ibukiyama's conjectures, we conjecture the automorphic origin and the R(π)-sign of every eigenvalue: Eisenstein, Saito-Kurokawa, Yoshida and paramodular eigenvalues, and pairs of type-Va eigensystems that only R(π) separates. We compute Gp at 33 primes, with up to 2176 vertices, and certify the predicted eigenvalues and signs in exact arithmetic; at p=167 and $173$, where the sign prediction first becomes non-trivial, the graph carries the predicted eigenvalue −8 with sign −1. For every prime we prove that each type-Va parameter contributes exactly one eigenline of each R(π)-sign, a closed formula for the trace of R(π), and a Fricke sign bias. If no further CAP representations occur, the non-trivial spectrum lies in [−8√2, 6+4√2], both ends being sharp under the conjecture, and only Saito-Kurokawa eigenvalues exceed 8√2; at all 104 primes p≤ 601 the extreme eigenvalue computed by Florit and Smith agrees with the largest Saito-Kurokawa one. Finally, the irrational roots of the residual square factor form one Galois orbit with full symmetric Galois group at each certified prime p≥ 29, and the spectrum appears to follow a Plancherel law.
Related
BibTeX
@misc{dang2026richelot,
title = {{The Richelot isogeny graph and paramodular forms of weight 3: Atkin-Lehner signs and certified spectra}},
author = {Hung T. Dang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.14145},
eprint = {2608.14145},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.14145},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Richelot isogeny graph and paramodular forms of weight 3: Atkin-Lehner signs and certified spectra
AU - Hung T. Dang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2608.14145
UR - https://arxiv.org/abs/2608.14145
AB - The Richelot (2,2)-isogeny graph G_p of superspecial abelian surfaces in characteristic p is the genus-2 analogue of the supersingular isogeny graph: its weighted adjacency matrix is a Brandt matrix for a compact inner form of GSp_4, and it commutes with an Atkin-Lehner involution R(π). Refining Ibukiyama's conjectures, we conjecture the automorphic origin and the R(π)-sign of every eigenvalue: Eisenstein, Saito-Kurokawa, Yoshida and paramodular eigenvalues, and pairs of type-Va eigensystems that only R(π) separates. We compute G_p at 33 primes, with up to 2176 vertices, and certify the predicted eigenvalues and signs in exact arithmetic; at p=167 and $173$, where the sign prediction first becomes non-trivial, the graph carries the predicted eigenvalue −8 with sign −1. For every prime we prove that each type-Va parameter contributes exactly one eigenline of each R(π)-sign, a closed formula for the trace of R(π), and a Fricke sign bias. If no further CAP representations occur, the non-trivial spectrum lies in [−8√2, 6+4√2], both ends being sharp under the conjecture, and only Saito-Kurokawa eigenvalues exceed 8√2; at all 104 primes p≤ 601 the extreme eigenvalue computed by Florit and Smith agrees with the largest Saito-Kurokawa one. Finally, the irrational roots of the residual square factor form one Galois orbit with full symmetric Galois group at each certified prime p≥ 29, and the spectrum appears to follow a Plancherel law.
ER -