Sampling from $p$-adic algebraic manifolds

TL;DR

This paper introduces a novel method for sampling points from algebraic manifolds over local fields, using random linear slices, with applications to p-adic groups and algebraic varieties.

Contribution

It extends sampling techniques to p-adic algebraic manifolds, providing an implementation and demonstrating practical applications in various algebraic structures.

Findings

Effective sampling from p-adic algebraic groups and varieties

Implementation demonstrates applicability across multiple contexts

Method generalizes real algebraic manifold sampling techniques

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.