Non-Abelian Cohomological Obstructions and the Asymptotic Spectral Gap of Supersingular Isogeny Graphs over Higher-Dimensional Abelian Varieties

The imminent advent of cryptographically relevant quantum computers hasprecipitated a paradigm shift in algebraic geometry and number-theoretic cryptography,necessitating the exploration of post-quantum primitives resistant to Shor’s algorithm.While supersingular isogeny graphs of elliptic curves (dimension g = 1) have been exten-sively studied, recent cryptanalytic breakthroughs—specifically the Castryck-Decru attackon SIDH—have exposed the fragility of auxiliary torsion information in low-dimensionalmoduli spaces. This paper presents a monumental and exhaustive investigation intothe asymptotic spectral properties of supersingular isogeny graphs Gg,p associated withprincipally polarized abelian varieties of dimension g ≥ 2 over finite fields Fp2 . Werigorously formulate a theoretical framework identifying non-abelian cohomological ob-structions in H1(Gal( ¯Fp/Fp), Aut(A)) that impede the free mixing of random walkson these high-dimensional expander graphs. By generalizing the Eichler-Deuring massformula to the Siegel modular variety Ag,1 and utilizing the trace formula for Heckeoperators on spaces of automorphic forms, we derive explicit bounds for the second largesteigenvalue λ2 of the adjacency operator. Our analysis reveals a paradoxical "spectralstratification" where the expansion constant (Cheeger constant) improves with dimension,yet the computational path-finding problem becomes exponentially harder due to theproliferation of cohomological barriers preventing efficient lifting of isogeny cycles. Weprovide extensive numerical data for g = 2 and g = 3, simulating the spectral gapevolution over large primes, and demonstrate that the non-abelian obstructions induce alocalized clustering effect that defies classical Ramanujan graph predictions. This workestablishes a new foundation for high-dimensional isogeny-based cryptography, proposinga modified protocol that leverages these obstructions as a security feature rather than avulnerability.