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Smoothness of Shortest Isogenies to Frobenius Conjugates
Abstract
Let E be chosen uniformly from the geometric isomorphism classes of supersingular elliptic curves over 𝔽̅p and δE be the minimum degree of isogenies from E to its Frobenius conjugate E(p). We establish bounds for the smoothness probability of δE and hence affirm Wesolowski's p1/3+o(1) attack on the supersingular endomorphism ring problem is unconditional. We also establish a uniform asymptotic formula, with a main term expressed in terms of Dirichlet L values, for the probability that a connection isogeny of degree n exists between E and E(p).
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BibTeX
@misc{ouyang2026smoothness,
title = {{Smoothness of Shortest Isogenies to Frobenius Conjugates}},
author = {Yi Ouyang and Wei Wei and Zheng Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21488},
eprint = {2610.21488},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21488},
}
RIS
TY - UNPB
TI - Smoothness of Shortest Isogenies to Frobenius Conjugates
AU - Yi Ouyang
AU - Wei Wei
AU - Zheng Xu
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.21488
UR - https://arxiv.org/abs/2610.21488
AB - Let E be chosen uniformly from the geometric isomorphism classes of supersingular elliptic curves over 𝔽̅_p and δ_E be the minimum degree of isogenies from E to its Frobenius conjugate E^(p). We establish bounds for the smoothness probability of δ_E and hence affirm Wesolowski's p^1/3+o(1) attack on the supersingular endomorphism ring problem is unconditional. We also establish a uniform asymptotic formula, with a main term expressed in terms of Dirichlet L values, for the probability that a connection isogeny of degree n exists between E and E^(p).
ER -