Newton strata realization for hypersurfaces via explicit p-adic cohomology

TL;DR

This paper develops optimized algorithms for computing zeta functions of hypersurfaces over finite fields using p-adic cohomology, achieving state-of-the-art performance and new explicit examples.

Contribution

It introduces new reduction policies and GPU-optimized implementation for Kedlaya's algorithm, enabling efficient zeta function computations for complex hypersurfaces.

Findings

Achieved faster zeta function computations for quintic curves and cubic fourfolds.

First systematic computations of zeta functions of quintic surfaces.

Generated new explicit examples of varieties with diverse Newton polygons.

Abstract

Let $X$ be a smooth projective hypersurface over a finite field $k$ of characteristic $p$. We address the problem of practically computing the zeta function $Z(X,T)$ of $X$ (equivalently, the point counts $\#X(\mathbb{F}_q)$, where $q = p^n$), and we focus on the case when $7 \leq p < 50$. We use the theoretical framework of the variant of Kedlaya's algorithm in arXiv:archive/0601508, and we use the technique of controlled reduction as described in Costa's Thesis. We define an optimization problem that abstracts the key bottleneck in the implementation of controlled reduction. An algorithm that solves this problem is called a reduction policy. We present three reduction policies with different advantages and disadvantages. We also present a high-performance implementation of controlled reduction that contains GPU-optimized linear algebra code and a data structure for linear recurrences that the authors hope can be used to study further reduction policies. Our algorithms get state-of-the-art performance in many cases; for example, we beat arXiv:1402.6758 or arXiv:2203.02070 on many examples of quintic curves, while also being able to compute zeta functions of cubic fourfolds when $p = 7$. We also have the first (to our knowledge) systematic computations of zeta functions of quintic surfaces. We use our implementation to deduce many new explicit examples of varieties with specified Newton polygons, including a cubic fourfold which are neither ordinary nor supersingular, quartic K3 surfaces of various Artin-Mazur heights, and quintic surfaces of all possible domino numbers.