The Richelot isogeny graph and paramodular forms of weight 3: Atkin-Lehner signs and certified spectra

Abstract

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 ${\mathrm{GSp}}_4$, and it commutes with an Atkin-Lehner involution $R(\pi )$. Refining Ibukiyama's conjectures, we conjecture the automorphic origin and the $R(\pi )$-sign of every eigenvalue: Eisenstein, Saito-Kurokawa, Yoshida and paramodular eigenvalues, and pairs of type-Va eigensystems that only $R(\pi )$ 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 $R(\pi )$-sign, a closed formula for the trace of $R(\pi )$, and a Fricke sign bias. If no further CAP representations occur, the non-trivial spectrum lies in $[-8\sqrt{2},6+4\sqrt{2}]$, both ends being sharp under the conjecture, and only Saito-Kurokawa eigenvalues exceed $8\sqrt{2}$; at all 104 primes $p\le 601$ 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 $p\ge 29$, and the spectrum appears to follow a Plancherel law.

arXiv comment41 pages. v3: retitled; certified range extended to p=167,173, the first primes with a Fricke-plus non-lift (predicted sign -1 confirmed); general-type rows proved for every prime; spectral bound [-8 sqrt 2, 6+4 sqrt 2], tested against Florit-Smith up to p=601; Plancherel law; corrections to (6) at p=113 and to the trace-identity proof. Data and code: doi:10.5281/zenodo.20798967