A structural trace identity and certified spectra for the Richelot-Brandt graph

Abstract

The degree-$2$ Brandt operator $B_2(2)$ on the principal genus of binary quaternion Hermitian lattices of discriminant $p$ is the weighted adjacency operator of the Richelot $(2,2)$-isogeny graph on superspecial principally polarized abelian surfaces, and commutes with an Atkin-Lehner involution $R(\pi)$. For every prime $p\ge7$ we prove that the trace of $R(\pi)$ is the sum of an explicit lift contribution from elliptic newforms of weights $2$ and $4$ and a signed defect of the weight-$3$ paramodular non-lift space; the closed formula for the defect yields the Fricke-sign bias $d(p)\ge0$ for every prime. We formulate an eigenvalue-sign refinement of Ibukiyama's principal-genus multiplicity conjectures: charpoly $B_2(2)$ factors into Eisenstein, Saito-Kurokawa, opposite-sign Yoshida, type-Va, and general-type blocks with specified $R(\pi)$-signs. The type-Va clause is a theorem for every prime: by the global lifting theorem of Roesner and Weissauer for inner forms anisotropic at the archimedean place, the weak packet of a general-type representation of $GU_2(B)$ is the full product of its local $L$-packets, each member occurring with multiplicity one, so both members of every type-Va pair occur and the type-Va block is an exact square split evenly by $R(\pi)$. For the Saito-Kurokawa and Yoshida blocks the refinement remains conjectural. Exact-arithmetic certificates, replayable from a frozen archive, verify the full prediction at every prime $11\le p\le149$: at $p=19$ the first type-Va pair is separated by $R(\pi)$, and at $p=61$ the graph realizes the general-type factor $x+7$.