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Sampling from $p$-adic algebraic manifolds

Yassine El Maazouz, Enis Kaya

TL;DR

This work introduces a non-archimedean, $p$-adic analogue of random-point sampling on algebraic manifolds by intersecting the target variety with random affine subspaces of complementary dimension. The core idea leverages the weight function $w_X$ and the $p$-adic co-area formula to convert integrals into expectations and to produce exact density-preserving samples from $X$ with a prescribed density $f$. The authors prove affine and projective versions of the main theorems, provide practical sampling procedures for linear spaces, and supply a SageMath implementation. The framework is demonstrated on concrete contexts, including measures on algebraic groups and moduli spaces such as modular curves and Hilbert modular surfaces, illustrating its potential for arithmetic statistics and computational algebraic geometry in the $p$-adic setting.

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 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.

Sampling from $p$-adic algebraic manifolds

TL;DR

This work introduces a non-archimedean, -adic analogue of random-point sampling on algebraic manifolds by intersecting the target variety with random affine subspaces of complementary dimension. The core idea leverages the weight function and the -adic co-area formula to convert integrals into expectations and to produce exact density-preserving samples from with a prescribed density . The authors prove affine and projective versions of the main theorems, provide practical sampling procedures for linear spaces, and supply a SageMath implementation. The framework is demonstrated on concrete contexts, including measures on algebraic groups and moduli spaces such as modular curves and Hilbert modular surfaces, illustrating its potential for arithmetic statistics and computational algebraic geometry in the -adic setting.

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 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 -adic algebraic groups, abelian varieties, and modular curves.
Paper Structure (22 sections, 16 theorems, 136 equations, 3 figures, 2 tables)

This paper contains 22 sections, 16 theorems, 136 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The setup
  3. Main results
  4. Notation
  5. Background and preliminaries
  6. A primer on non-archimedean local fields
  7. Norms, Lattices, and Gaussian Measures
  8. The weight function of an affine algebraic manifold
  9. A pinch of intersection theory
  10. The -adic co-area formula
  11. Sampling from affine manifolds
  12. Sampling from projective manifolds
  13. Preliminaries
  14. Sampling from projective manifolds
  15. Sampling linear spaces in practice
  16. ...and 7 more sections

Key Result

Theorem 1.2

Let $X \subset \mathbb{A}^{N}$ be an $n$-dimensional affine algebraic manifold defined over $K$. Let $(\bm{A},\bm{b})$ be a random variable in $K^{n \times N} \times K^n$ with independent entries uniformly distributed in $\mathcal{O}$, i.e., $(\bm{A},\bm{b})$ have distribution $1_{A \in \mathcal{O}^

Figures (3)

  • Figure 1: An illustration of the sampling method. The dotted lines are rejected; the red line intersects in 3 points the curve from which we wish to sample. A point is randomly sampled from the three points and the selected point is colored in green.
  • Figure 2: An illustration of the $3$-adic ring $\mathbb{Z}_3$. This picture shows all of the balls in $\mathbb{Z}_3$ of radius $r$ such that $3^{-2} \leq r \leq 1$. These are of the form $a + 3^k \mathbb{Z}_3$ where $0 \leq k \leq 2$ and $a \in \{ 0, 1, \dots, 3^{2} - 1 \}$. Because of the ultrametric nature of the $3$-adic absolute value, any two balls of the same radius are either equal or disjoint. This is a feature of the $p$-adic topology.
  • Figure 3: Minimal skeleta of the Berkovich analytification of genus-$2$ curves.

Theorems & Definitions (49)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Example 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 39 more