A uniform bound for the minimal degree of an isogeny between principally polarized superspecial abelian surfaces
Abstract
Let p>3 be a prime number. If A and B are two principally polarized superspecial abelian surfaces over 𝔽p2 with p2-Frobenius [−p], we prove that there exists a separable polarized isogeny between them with multiplier at most p/√2. The multiplier bound is uniform in the pair and asymptotically optimal up to a small multiplicative constant. Given quaternionic coordinates, we provide a deterministic algorithm that satisfies this upper bound in polynomial time. For unrestricted multipliers, KLPT2 provides a heuristic algorithm with a multiplier upper bound of order p6+o(1) for arbitrary pairs and p3+o(1) when one polarization matrix is the identity matrix. Our algorithm's improvement is an explicit upper bound below p with an unconditional deterministic guarantee.
Related
- Small degree isogenies between conjugate principally polarized superspecial abelian surfaces
- Existence of Ulrich bundles on polarized surfaces and weakly Ulrich bundles on polarized varieties in positive characteristic
- Some applications of the moduli space of U⊕ D8(−1)-polarized K3 surfaces
- On the classification of polarized surfaces with weakly Ulrich or initialized aCM tangent bundles
- Syzygies of Polarized Abelian Surfaces: A Reider-Type Criterion
BibTeX
@misc{pham2026uniform,
title = {{A uniform bound for the minimal degree of an isogeny between principally polarized superspecial abelian surfaces}},
author = {Lam L. Pham},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12080},
eprint = {2610.12080},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12080},
}
RIS
TY - UNPB
TI - A uniform bound for the minimal degree of an isogeny between principally polarized superspecial abelian surfaces
AU - Lam L. Pham
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12080
UR - https://arxiv.org/abs/2610.12080
AB - Let p>3 be a prime number. If A and B are two principally polarized superspecial abelian surfaces over 𝔽_p^2 with p^2-Frobenius [−p], we prove that there exists a separable polarized isogeny between them with multiplier at most p/√2. The multiplier bound is uniform in the pair and asymptotically optimal up to a small multiplicative constant. Given quaternionic coordinates, we provide a deterministic algorithm that satisfies this upper bound in polynomial time. For unrestricted multipliers, KLPT^2 provides a heuristic algorithm with a multiplier upper bound of order p^6+o(1) for arbitrary pairs and p^3+o(1) when one polarization matrix is the identity matrix. Our algorithm's improvement is an explicit upper bound below p with an unconditional deterministic guarantee.
ER -