Sampling from \(\boldsymbol{p}\)-Adic Algebraic Manifolds
Abstract.
We present a method for sampling points from an algebraic manifold, either affine or projective, defined over a local field, with a prescribed probability distribution. Inspired by the work of Breiding and Marigliano [SIAM J. Math. Data Sci., 2 (2020), pp. 683–704] on sampling real algebraic manifolds, our approach leverages slicing the given variety with random linear spaces of complementary dimension. We also provide an implementation of this sampling technique and demonstrate its applicability to various contexts, including sampling from linear \(p\)-adic algebraic groups, abelian varieties, and modular curves.
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Acknowledgments.
This work was initiated during the first author’s visit to the Max Planck Institute for Mathematics in the Sciences. We are grateful to the institute for its warm hospitality and inspiring research environment. We extend our thanks to Peter Bürgisser, Avinash Kulkarni, and Antonio Lerario for kindly sharing an early draft of their manuscript [9]. Special thanks go to Corrine Elliott, Steven N. Evans, Kiran Kedlaya, Bartosz Naskrecki, Bernd Sturmfels, and Tristan Vaccon for their insightful discussions and valuable feedback. Finally, we thank the anonymous referees for their thoughtful suggestions, which greatly improved the presentation and readability of this manuscript.
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Published In
SIAM Journal on Applied Algebra and Geometry
Pages: 343 - 373
DOI: 10.1137/24M163774X
ISSN (online): 2470-6566
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© 2025 Society for Industrial and Applied Mathematics.
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Submitted: 7 February 2024
Accepted: 19 February 2025
Published online: 14 May 2025
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