Principal-genus multiplicities beyond the certified range

Prove the parameter-by-parameter multiplicity assertion of Conjecture C1 for the principal genus of binary quaternion Hermitian lattices for every prime p>149, including the predicted multiplicity two for type-Va parameters and the splitting of each type-Va pair into one R(π)-eigenline of sign +1 and one of sign −1.

Background

Conjecture C1 predicts a complete factorization of the characteristic polynomial of the degree-2 Brandt operator B_2(2) on the principal genus, together with the signs of the commuting Atkin–Lehner involution R(π). In particular, it predicts that every type-Va pair occurs with multiplicity two and is separated by opposite R(π)-signs.

The paper verifies these predictions by exact-arithmetic certificates for primes 11≤p≤149. It proves the structural signed trace for all relevant primes, but explicitly notes that this trace does not determine the individual automorphic parameters or their multiplicities. A general representation-theoretic proof beyond the certified range remains unresolved.

References

Nothing in (i) or (ii) implies (iii) beyond the range certified by (iv); the multiplicity statement for $p>149$ remains open.

— A structural trace identity and certified spectra for the Richelot-Brandt graph  (2608.14145 - Dang, 14 Aug 2026) in Section 1, subsection “The problem”; Section 2, subsection “The certified finite verification”; Section 7, subsection “Open problems,” item (2)