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Isogeny graphs on superspecial abelian varieties: eigenvalues and connection to Bruhat–Tits buildings
Published online by Cambridge University Press: 20 October 2023
Abstract
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.
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society
Footnotes
Y.A. is supported by JST, ACT-X Grant Number JPMJAX2001, Japan. R.T. is partially supported by JSPS Grant-in-Aid for Scientific Research JP20K03602 and JST, ACT-X Grant Number JPMJAX190J, Japan. T.Y. is partially supported by JSPS KAKENHI Grant Number (B) 19H01778.
References
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