Hyperelliptic Curves over Small Finite Fields and GPU Accelerators

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.