Geodesic
Isogeny graphs on superspecial abelian varieties: eigenvalues and connection to Bruhat–Tits buildings
Canadian journal of mathematics, Tome 76 (2024) no. 6, pp. 1891-1916

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We study for each fixed integer $g \ge 2$, for all primes $\ell $ and p with $\ell \neq p$, finite regular directed graphs associated with the set of equivalence classes of $\ell $-marked principally polarized superspecial abelian varieties of dimension g in characteristic p, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of p. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles–Goren–Lauter-type cryptographic hash functions for abelian varieties. We give explicit lower bounds for the gaps in terms of the Kazhdan constant for the symplectic group when $g \ge 2$. As a byproduct, we also show that the finite regular directed graphs constructed by Jordan and Zaytman also has the same property.
DOI : 10.4153/S0008414X23000676
Mots-clés : Isogeny graphs, cryptographic hash functions, superspecial abelian varieties, isogeny graphs, Bruhat—Tits buildings 
Aikawa, Yusuke; Tanaka, Ryokichi; Yamauchi, Takuya. Isogeny graphs on superspecial abelian varieties: eigenvalues and connection to Bruhat–Tits buildings. Canadian journal of mathematics, Tome 76 (2024) no. 6, pp. 1891-1916. doi: 10.4153/S0008414X23000676
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     title = {Isogeny graphs on superspecial abelian varieties: eigenvalues and connection to {Bruhat{\textendash}Tits} buildings},
     journal = {Canadian journal of mathematics},
     pages = {1891--1916},
     year = {2024},
     volume = {76},
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     doi = {10.4153/S0008414X23000676},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X23000676/}
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