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Isogeny graphs on superspecial abelian varieties: eigenvalues and connection to Bruhat–Tits buildings
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 6 / December 2024
- Published online by Cambridge University Press:
- 20 October 2023, pp. 1891-1916
- Print publication:
- December 2024
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- Article
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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.
