Hyperelliptic Curves over Small Finite Fields and GPU Accelerators

TL;DR

This paper introduces a GPU-accelerated method for computing Hasse-Weil invariants of hyperelliptic curves over small finite fields, enhancing algebraic computations by leveraging modern hardware for improved performance.

Contribution

It presents a novel GPU-based approach for algebraic computations on hyperelliptic curves, specifically focusing on Hasse-Weil invariants and Zech arithmetic performance analysis.

Findings

GPU acceleration significantly improves computation speed

Lookup table based Zech arithmetic is effective on GPUs

Method facilitates large-scale algebraic computations over finite fields

Abstract

We present a hardware-accelerated computation of Hasse-Weil invariants of all hyperelliptic curves of given genus over a fixed finite field. Our main motivation is the determination of traces of Frobenius on cohomology corresponding moduli stacks \`a la Bergstr\"om-Faber-van-der-Geer. This paper also constitutes a case study of the performance of lookup table based Zech arithmetic on Graphics Processing Units (GPUs). GPUs are by now ubiquitous in numerics, to an extend that their development itself is propelled by scientific computing. Algebraic computing has profited very little from the advancement of hardware design. We suggest that specific computations can be benificially adjusted to GPUs with comparatively little effort.